aa r X i v : . [ m a t h . P R ] M a r PATTERNS IN RANDOM FRACTALS
PABLO SHMERKIN AND VILLE SUOMALA
Abstract.
We characterize the existence of certain geometric configurations inthe fractal percolation limit set A in terms of the almost sure dimension of A .Some examples of the configurations we study are: homothetic copies of finitesets, angles, distances, and volumes of simplices. In the spirit of relative Szemer´editheorems for random discrete sets, we also consider the corresponding problem forsets of positive ν -measure, where ν is the natural measure on A . In both caseswe identify the dimension threshold for each class of configurations. These resultsare obtained by investigating the intersections of the products of m independentrealizations of A with transversal planes and, more generally, algebraic varieties,and extend some well known features of independent percolation on trees to asetting with long-range dependencies. Contents
1. Introduction and summary of main results 21.1. Introduction 21.2. Summary of results 41.3. General strategy 82. Notation 93. Preliminaries on fractal percolation 124. A class of random measures and their intersections with parametrizedfamilies of deterministic measures 154.1. Random measures 164.2. Martingale condition 174.3. Spatial independence 174.4. H¨older continuity of intersections 19
Mathematics Subject Classification.
Primary: 05D40, 28A75, 60C05; Secondary: 05C55,28A78, 28A80, 60D05.
Key words and phrases. configurations, progressions, random measures, random sets, martin-gales, Hausdorff dimension, fractal percolation, intersections.P.S. was partially supported by Projects PICT 2013-1393 and PICT PICT 2014-1480 (AN-PCyT). Part of this research was completed while P.S. was visiting the University of Oulu.V.S. was partially supported by the Academy of Finland via the Centre of Excellence in Analysisand Dynamics Research.
5. Affine intersections and linear patterns 265.1. Intersections with affine planes 265.2. Finite patterns in fractal percolation 355.3. Optimality of the results 386. Nonlinear intersections and applications 416.1. Preliminaries 416.2. Continuity for intersections with algebraic varieties 446.3. Non-linear configurations 496.4. Scale-invariant patterns 537. Patterns in sets of positive measure 557.1. The dimension of intersections and patterns 557.2. Lack of patterns in sets of full measure 587.3. Patterns in sets of positive measure 60References 641.
Introduction and summary of main results
Introduction.
A classical general problem in combinatorics is to understandwhat conditions (especially, conditions of structure and size ) on a set A imply that A contains certain configurations, like 3-term arithmetic progressions. Indeed, theclassical theorem of Roth [32] implies that if A ⊂ N has positive upper density, thenit must contain 3-term arithmetic progressions. The famous theorem of Szem´erdi[38] generalizes this to arbitrarily long arithmetic progressions. The celebratedtheorem of Green and Tao [13] generalizes the former statement to subsets of theprimes, and has stimulated a large amount of further research over the past decade.There has been much interest also in this kind of problems when A is a subsetof Euclidean space. One might heuristically conjecture that if A ⊂ R is ‘large’,then it should contain progressions. If A is large in the sense of measure, thenthis is indeed the case. A well known corollary of the Lebesgue density theoremasserts that any set A ⊂ R with positive Lebesgue measure contains arbitrary longarithmetic progressions and, more generally, in any dimension, homothetic copiesof all finite sets. A wide open conjecture of Erd˝os states that for all infinite sets S ⊂ R , there is a set A ⊂ R with positive measure which does not contain a similarcopy of S .In the zero-measure case, a natural candidate for size is Hausdorff dimension.Already in 1959, Davies, Marstrand, and Taylor [6] showed that there are compactsets of dimension 0 which contain a homothetic copy of all finite sets. This wasrecently extended to polynomial patterns in [29]. Hence, a small dimension itselfdoes not rule out containing rich sets of configurations. On the other hand, there are ATTERNS IN RANDOM FRACTALS 3 compact sets A ⊂ R of Hausdorff dimension 1 without any arithmetic progressions(see [21]) and compact sets in arbitrary dimension that do not contain parallelograms[26]. Thus, Hausdorff dimension of A ⊂ R alone gives no information whatsoeverabout the existence of arithmetic progressions (and certain other configurations) in A . We note, however, that the situation is different for other patterns. For example,Iosevich and Liu [18] recently proved that for d ≥ ε d > A of R d of dimension > d − ε d contains the vertices of an equilateraltriangle. This was known to be false in dimension d = 2 ([9, 26]) and the problemis open in dimension d = 3.It turns out that in many cases a lower bound on the Hausdorff dimension doesimply the presence of a positive measure of configurations in some class (even if itoften does not guarantee the existence of any one given configuration). Perhapsthe most classical example is the distance set problem: Falconer [8] conjecturedthat if A ⊂ R d , d ≥ H ( A ) > d , then its distance set D ( A ) = {| x − y | : x, y ∈ A } has positive Lebesgue measure. The best current resultstowards this conjecture are due to Wolff [39] and Erdogan [7]: dim H ( A ) > d + suffices. Many other problems of a similar kind have been investigated, see e.g.[14, 12] and references there.A fruitful parallel line of work has focused on finding pseudo-randomness condi-tions (in addition to size conditions) on subsets of R d that ensure the presence of con-figurations (such as arithmetic progressions). Typically, these pseudo-randomnessconditions take the form of a suitable Fourier decay of a measure supported onthe set in question: see [23, 1, 2, 17]. Recall that the Fourier transform of a finitemeasure µ on R d is given by ˆ µ ( ξ ) = R exp( − πiξ · x ) dµ ( x ). To a give a flavourfor this type of results, we state a very special case of [17, Theorem 1.3]: given D > , β ∈ (0 , ε ( D, β ) > µ is ameasure on R such that µ ([ x − r, x + r ]) ≤ Dr α for all r > , x ∈ R , ˆ µ ( k ) ≤ D (1 − α ) − B | k | − β/ for all k ∈ N , then, provided that α > − ε ( D, β ), the topological support of µ contains a 3-termarithmetic progression. In fact, the results from [17] apply to many linear, and somepolynomial, patterns in R d . An interesting feature of [1] is that the progressionscan be found in any set of positive µ -measure, not just the topological support.Although these are deep results, we note that the hypotheses are difficult to verifyfor concrete examples, in part because the bound on the mass decay exponent α depends on the constant D (it was shown in [34] that such dependence cannot beremoved). The examples showing that measures satisfying conditions such as the PABLO SHMERKIN AND VILLE SUOMALA above exist are random, see [23]. Moreover, the value of ε ( D, β ) is not explicit, andcertainly far from sharp.Going back to the discrete setting, the last few years saw an explosion of relative
Szemer´edi Theorems. That is, given some discrete set A , one is interested in knowingwhether sets of positive density relative to A contain large arithmetic progressions.The Green-Tao Theorem mentioned above is of this type, with A equal to the primenumbers. A general approach to relative Szemer´edi theorems (which in particularyields a simpler proof of the Green-Tao Theorem) was developed in [5]. Closerto our work, sharp relative Szemer´edi theorems have been obtained for randomdiscrete sets by Conlon and Gowers [4] and, independently, by Schacht [33]: for δ > k ∈ N ≥ , let us say that a set A ⊂ { , . . . , n } is ( δ, k )-Szemer´edi if everysubset A ′ ⊂ A with | A ′ | ≥ δ | A | contains an arithmetic progression of length k . If [ n ] p denotes the canonical random set obtained by keeping each number in { , . . . , n } independently with probability p then, provided p ≥ Cn − / ( k − (with C a suitablylarge absolute constant), the probability that [ n ] p is ( δ, k )-Szemer´edi tends to 1 as n → ∞ . Moreover, this threshold is sharp (up to the value of C ). We note that thethreshold for the existence of k -progressions in A itself is p ∼ n − /k , and this is afar more elementary fact.1.2. Summary of results.
This circle of results show that, despite the substantialprogress achieved, the connection between size, pseudo-randomness and the pres-ence of progressions and other patterns, is far from being elucidated, especially inthe continuous setting. The goal of this work is to present a systematic study of theexistence of patterns in random fractals. That is, rather than dealing with pseudo-randomness (such as fast Fourier decay), we will consider ‘honest’ random sets andmeasures. This will also give us the chance to explore ‘relative Szemer´edi’ type ofresults in our setting.Unlike the discrete case, there is no canonical random set or measure of fractionaldimension. In [36], we proposed a large class of random fractal measures on Eu-clidean space which aims to capture the main properties of the canonical discreterandom set. For concreteness, in this article we focus on what is perhaps the bestknown and studied model of stochastically self-similar set: fractal percolation. Nev-ertheless, the method should work for many other random fractals, including farmore general subdivision constructions and Poissonian cutouts. In fact, our mainabstract result in Section 4.4 holds for a wide variety of random measures satisfyingsuitable martingale and weak dependency conditions.In order to state some of our results more precisely, let us recall the definition offractal percolation. For convenience of notation, we will consider fractal percolationon the dyadic grid only. Fix a parameter p ∈ (0 ,
1) and d ∈ N . We subdivide theunit cube in R d into 2 d equal sub-cubes. We retain each of them with probability p ATTERNS IN RANDOM FRACTALS 5 and discard it with probability 1 − p , with all the choices independent. For each ofthe retained cubes, we continue inductively in the same fashion, further subdividingthem into 2 d equal sub-cubes, retaining them with probability p and discardingthem otherwise, with all the choices independent of each other and the previoussteps. The fractal percolation limit set A = A perc( d,p ) is the set of points which arekept at each stage of the construction. It is well known that if p ≤ − d , then A isalmost surely empty, and otherwise a.s.dim H A = dim B A = s ( d, p ) := d + log p (1.1)conditioned on non-extinction (i.e. A = ∅ ). Here, and throughout the paper,dim H , dim B denote Hausdorff and box-counting (Minkowski) dimensions, respec-tively. Fractal percolation can be seen as a Euclidean realization of a Galton-Watsonbranching process. See [25] for extensive background on branching processes, fractalpercolation and dimension.Our first class of results identify the dimension threshold s ( d, p ) for the presenceof a wide variety of geometric configurations in A : Theorem 1.1.
The following hold for A = A perc ( d,p ) , provided the required conditionson d and s = s ( d, p ) hold:(1) If m ≥ and s > d − ( d + 1) /m , then A contains a homothetic copy of all m -point sets.(2) If m ≥ and s > d − d/m , then for any subset { x , . . . , x m } ⊂ ]0 , d , A contains a translation of { x ′ , . . . , x ′ m } whenever x ′ i are close enough to x i .(3) If d ≥ and s > / , then there is ε > such that (0 , ε ) ⊂ D ( A ) .(4) If s > / ( d + 1) , then there is ε > such that A contains the vertices of asimplex of all volumes in (0 , ε ) .(5) If d = 2 and s > , then for { x , x , x } ⊂ ]0 , , A contains an isometriccopy of { x ′ , x ′ , x ′ } whenever x ′ i are close enough to x i .(6) If d ≥ and s > / , then triples of points in A determine all angles in ]0 , π [ .(7) If d ≥ and s > / , then A contains the vertices of all non-degeneratetriangles, up to similarities.(8) If m ≥ , d = 2 and s > − /m , then up to similarities A contains thevertices of all non-degenerate m -gons.To be more precise, the claims (2) and (5) hold with positive probability and theothers hold a.s. on A = ∅ . Moreover, in all these cases, the range of s is sharp, inthe sense that if s is smaller or equal than the given threshold, then any one givenconfiguration has probability zero of occurring in A . For example, for any m -pointset S ⊂ R d , if s ≤ d − ( d + 1) /m , then a.s. A contains no similar copy of S . PABLO SHMERKIN AND VILLE SUOMALA
Furthermore, the thresholds are sharp for packing dimension (up to the endpoint),even for deterministic sets. For example, if A ⊂ R d contains a homothetic copy ofall m -point sets, then dim P ( A ) ≥ d − ( d + 1) /m . This theorem will be proved in the course of Sections 5–6. For now, we makesome general remarks:
Remarks . (i) Proving the existence of a single configuration is already morechallenging than in the random discrete setting, although in general it canbe done via the second moment method (see e.g. Lemma 5.7). However,the main challenge is proving the existence of open sets/all configurationssimultaneously, which is an issue that obviously does not arise in the discreteworld.(ii) All the configurations arising in Theorem 7.1 can be realized as the zero set ofa suitable polynomial, and the dimension thresholds are derived from a generalstatement about intersections (of the Cartesian powers of A ) with algebraicvarieties, see Corollary 6.8.(iii) The statement about the distance set of A was proved, in a slightly weakerform, by Rams and Simon [31]. Although we use some of their ideas (as we didalready in our paper [36]), there are substantial differences that allow us to getstronger results, including the ‘relative Szemer´edi’ version discussed below.One can visualize Theorem 1.1 by considering the following joint construction ofall fractal percolation processes. Let ( U Q ) be a sequence of IID random variables,uniform in [0 , Q ranges over all dyadic cubes of all levels, starting withthe unit cube. Given any p , we can construct a set A p by retaining cubes Q forwhich U Q ≤ p , and discarding those with U Q > p . In this way we get an increasingensemble ( A p ) p ∈ [0 , , where A p has the distribution of A perc( d,p ) . Theorem 1.1 thenshows that almost surely the sets A p undergo a phase transition for the presenceof geometric configurations at the corresponding critical value of p . For example,given a fixed m -element set S in R d , A p contains no homothetic copy of S as long aslog (1 /p ) ≥ d +1 m , but as soon as log (1 /p ) < d +1 m , the set A p transitions to containinga homothetic copy not just of S but of all m -point configurations.We are able to sharpen Theorem 1.1 as follows: for each class of configurations,if s is above the given threshold, not only we get that A contains all/an open setof configurations, but we can precisely measure how often each configuration arises.We give only one example here, deferring further discussion to Section 7. Theorem 1.3.
Let A = A perc ( d,p ) . If s = s ( d, p ) > d − d +1 m , then almost surely on A = ∅ , for each m -point S ⊂ R d , dim H ( { ( a, b ) ∈ (0 , ∞ ) × R d : aS + b ⊂ A } ) = m ( s − d ) + d + 1 . ATTERNS IN RANDOM FRACTALS 7
There is a natural random measure supported on the fractal percolation limitset (this is sometimes called ‘branching measure’ in the context of Galton-Watsontrees). This is obtained as the weak-* limit of the measures ν n := p − n L d | A n , where A n is the union of the surviving cubes of side-length 2 − n , and L d is d -dimensionalLebesgue measure (See Section 3 for more details). Then a.s. ν n converges weaklyto a limit ν = ν perc( d,p ) . Moreover, if p > − d , then ν = 0 a.s. on A = ∅ , and in thiscase the Hausdorff dimension of ν equals s ( d, p ) (that is, if A ′ is a Borel set with ν ( A ′ ) >
0, then dim H ( A ′ ) ≥ s ( d, p )).Positive ν -measure is then a natural analogue of ‘positive relative density’ inthe discrete random case, and this gives us a way to investigate relative Szemer´ediphenomena for fractal percolation: Theorem 1.4.
Let ν = ν perc ( d,p ) . Almost surely, the following holds for each Borelset A ′ such that ν ( A ′ ) > under the given conditions on d and s = s ( d, p ) :(1) If m ≥ and s > d − m − , then A ′ contains a homothetic copy of all m pointsets.(2) If s > , then the distance set of A ′ has non-empty interior.(3) If s > d , then the set of volumes of simplices with vertices in A ′ has non-empty interior.(4) If d = 2 and s > , then there is an open set of triples { a , a , a } such that A ′ contains an isometric image of { a , a , a } .(5) If d ≥ and s > , then A ′ contains all angles in ]0 , π [ .(6) If d ≥ and s > , then A ′ contains a similar copy of all non-degeneratetriangles.(7) If m ≥ , d = 2 and s > − m − , then A ′ contains a similar copy of allnon-degenerate m -gons.Moreover, these thresholds are sharp, in the sense that for any countable set ofconfigurations in each class, if s is smaller or equal than the given threshold, thena.s. there is a Borel set A ′ of full ν -measure which does not contain any configurationin the countable set. For example, if s ≤ , then there is a full ν -measure set A ′ which does not contain any rational distances. It is interesting to compare the different thresholds with what is known or con-jectured for deterministic sets. For example, we pointed out earlier that sets of fullHausdorff dimension in the line may fail to contain three-term progressions, and setsof full dimension in the plane may fail to contain equilateral triangles. Thus, (1)and (6) are very far from holding for general sets of the given dimensions. On theother hand, the distance set conjecture in the plane (but not in higher dimensions)almost gives (2) for any set A ′ of dimension > ν and the density PABLO SHMERKIN AND VILLE SUOMALA point theorem will allow us weaken the statement from ‘full probability’ to ‘positiveprobability’ and from ‘positive measure’ to ‘full measure’. On the other hand, asis already the case for Theorem 1.1, we have to deal with uncountable families ofconfigurations.1.3.
General strategy.
We will obtain all the aforementioned results by investi-gating the intersections of the Cartesian products A m ⊂ R md with families of affinesubspaces, and more general algebraic varieties. For instance, to show that A ⊂ R contains similar copies of all triples { t , t , t } ⊂ R , we have to show that A × A × A intersects the 2-dimensional plane V t = { ( x, x, x ) + λ ( t , t , t ) : x, λ ∈ R } ⊂ R , outside of the diagonal { x = y = z } , for all choices of t = ( t , t , t ). This willbe verified by considering the intersections or ‘slices’ of ν × ν × ν with the planes V t , and showing that the total mass of these intersections is bounded away fromzero. This will be achieved by showing that the total mass of a slice behaves in acontinuous way (as a function of t ). This continuity, in turn, will be derived as aconsequence of a general intersection result for weakly dependent martingales (tobe defined in Section 4.4 below). The main abstract result (Theorem 4.9) yieldsH¨older continuity for the map t Y t , where Y t is the total mass of the intersectionof µ and η t , where { η t } t ∈ Γ is a suitable deterministic family of measures on R d (orrather ( R d ) m in our applications), parametrized by a metric space Γ, and µ is arandom limit measure of absolutely continuous measures µ n satisfying certain sizeand ‘weak spatial dependence’ assumptions. This setup generalizes the spatiallyindependent martingales from [36]; in particular, Theorem 4.9 extends the mainresult of [36]. We hope this more general framework will find further applicationsbeyond those explored in this article. In a future work, we hope to relax also themartingale condition, and derive further applications to self-convolutions of ν andcertain maximal operators.To give an idea of the method, we discuss the proof of the existence of 3-patternsfor fractal percolation sets A ⊂ [0 , A is the weak limit of µ n = ν n × ν n × ν n , recall that ν n = p − n L| A n . Given t = ( t , t , t ) ∈ R , we considerthe total mass of the intersection of µ n and H | V t defined as Y tn = Z V t µ n ( x ) d H ( x ) . The increments Y tn +1 − Y tn may be expressed as sums of the random variables X Q = R Q ∩ V t µ n +1 ( x ) − µ n ( x ) d H ( x ) where Q runs over all dyadic cubes of side-length2 − n . If these random variables { X Q } were independent, and if µ n satisfied themartingale condition E ( µ n +1 ( x ) | A n ) = µ n ( x ) for all x ∈ R , n ∈ N , we could apply ATTERNS IN RANDOM FRACTALS 9 a large deviation estimate to show that Y tn converges very rapidly, and derive acontinuity modulus for the limit with respect to t (this is the strategy of [36], whichin turn was inspired in [30]). These assumptions would be satisfied, for instance, if µ n was the fractal percolation measure on [0 , , instead of the 3-fold self-productfractal percolation on [0 , µ n would be a model example of theSI-martingales considered in [36] and would allow us to conclude that the limits Y t = lim n →∞ Y tn are a.s. H¨older continuous in t , provided that the dimension ofthe limit measure µ is larger than 1. In the present situation, however, both themartingale condition and the spatial independence condition fail. For instance,if Q, Q ′ are two dyadic cubes with the same x -coordinate, then X Q and X Q ′ areclearly dependent. A priori, there can be many such dependencies, since the planes V t intersect the hyperplanes { x = c } in a line (and there could be many survivingcubes along this line). The martingale condition, on the other hand, breaks down atthe dyadic cubes meeting one of the diagonals { x = y } , { x = z } or { y = z } . It turnsout that the amount of dependencies can be inductively bounded by looking at theslices of the lower dimensional product ν n × ν n with ‘transversal’ lines. These boundsmake the dependencies sparse enough that a large deviation estimate for Y tn +1 − Y tn can still be derived, so that the continuity in t can then be established along thelines of [36], provided dim µ > ν = dim A > / e µ n = ν (1) n × ν (2) n × ν (3) n of threeindependent realizations, instead of µ n . Now e µ n is easily checked to be a martingale,although it has the same dependency issues as before. This strategy is formalizedin the general Theorem 4.9 below. 2. Notation
We will use Landau’s O ( · ) and related notation. If n > g ( n ) = O ( f ( n )) we mean that there exists C > ≤ g ( n ) ≤ Cf ( n )for all n . By g ( n ) = Ω( f ( n )) we mean f ( n ) = O ( g ( n )). Occasionally we willwant to emphasize the dependence of the constants implicit in the O ( · ) notationon other, previously defined, constants; the latter will be then added as subscripts.For example, g ( n ) = O δ ( f ( n )) means that 0 ≤ g ( n ) ≤ C δ f ( n ) for some constant C δ which is allowed to depend on δ .The notation B ( x, r ) stands for the closed ball with centre x and radius r . Openballs will be denoted by B ◦ ( x, r ). We will write E ( ε ) for the open ε -neighbourhood { x ∈ R M : dist( x, E ) < ε } . Moreover, E ◦ and E denote the interior and closure of E , respectively. Given L ∈ N , we let [ L ] = { , . . . , L } . We will denote by | · | boththe absolute value | x | of an element of R M , as well as the cardinality | I | of a (finite)set I . By a measure we always mean a locally finite Borel-regular outer measure ona metric space. Given a measure µ on R M , we denote k µ k = µ ( R M ). We willdenote by P the law of the fractal percolation as well as various other probabilitymeasures (it should be always clear from the context what probability measure weare referring to). In general, we will denote P -measurable events by C , F , etc.We denote by Q n (or Q Mn ) the family of dyadic cubes of R M with side length 2 − n ,and by Q the union ∪ n ∈ N Q n . It will be convenient that these are pairwise disjoint,so we consider a suitable half-open dyadic filtration.As noted earlier, dim H denotes Hausdorff dimension. We denote upper box-counting (or Minkowski) dimension by dim B , and box-counting dimension (when itexists) by dim B , while packing dimension is denoted by dim P . A good introductionto fractal dimensions can be found in [10, Chapters 2 and 3].The Grassmann manifold of k -dimensional linear subspaces of R M will be denoted G M,k . It is a compact manifold of dimension k ( M − k ), and its metric is d ( V, W ) = k π V − π W k , where π ( · ) denotes orthogonal projection. The manifold of k -dimensional affinesubspaces of R M will be denoted A M,k . It is diffeomorphic to G M,k × R M − k , andthis identification defines a natural metric. The metrics on all these different spaceswill be denoted by d ; the ambient space will always be clear from context (also notethat the ambient dimension is sometimes denoted by the same symbol d ).Starting from Section 5, we will be working on the space ( R d ) m for some integers m, d , which we sometimes shorten to R md . We will denote the elements of ( R d ) m by ( x , . . . , x m ), where x j = ( x j , . . . , x dj ) ∈ R d for each j ∈ [ m ], so that the (real)coordinates of x j are denoted x ij , i ∈ [ d ]. Given 1 ≤ j ≤ m , we will denote by π j the orthogonal projection onto the subspace H j := { x ∈ ( R d ) m : x i = 0 for i = j } ∈ G md,d . and by π j the projection onto the orthogonal complement of H j (which is an el-ement of G md, ( m − d ). Furthermore, we will identify each H j with R d , and H ⊥ j with ( R d ) m − . That is, for x = ( x , . . . , x m ) ∈ ( R d ) m , π j ( x ) = x j , π j ( x ) =( x , . . . , x j − , x j +1 , . . . , x m ). We will denote by ∆ ⊂ ( R d ) m the union of all thediagonals { x i = x j } , i = j . Furthermore, given an index set I ⊂ [ m ], we denote H I = { ( x , . . . , x m ) ∈ ( R d ) m : x j = 0 for all j ∈ I } . Moreover, if j ∈ [ m ] \ I and k ∈ [ d ], we let H I,j,k = { x ∈ H I : x kj = 0 } . In Sections 6 and 7, we will replace the linear subspaces V ∈ A M,k ( M = md )by algebraic varieties Z P = Z ( P ) := P − (0) ∩ [0 , M , where P is a polynomial ATTERNS IN RANDOM FRACTALS 11 L the Lebesgue measuredim H , dim P , dim B Hausdorff, packing, and box-counting dimensions( µ n ), µ a sequence of random measures and its (weak-*) limit k µ k the total mass of µA perc , ν perc fractal percolation set and the natural measure A n , ν n level n approximations of A = A perc and ν = ν perc N n the total number of cubes forming A n P the law of A n , ν n (or of some other random sequence µ n ) C , F P -measurable events DI n dependency degree { η t : t ∈ Γ } parametrized family of (deterministic) measures µ tn the ‘intersection’ of µ n and η t Y tn , Y t the total mass of µ tn , and its limit Q , Q M , the family of half-open dyadic cubes of R M Q n , Q Mn and the ones with side-length ℓ ( Q ) = 2 − n SIM M the family of non-singular similarities on R M G M,k the manifold of k -dimensional linear subspaces of R M A M,k the manifold of k -dimensional affine subspaces of R M V , W elements of A M,k P r,q,M = P r the polynomials P : R M → R q of degree ≤ r P reg r the regular polynomials in P r P, P , P elements of P r Z ( P ), Z P the set P − (0) ∩ [0 , M [ L ] the integers 1 , . . . , L ( x , . . . , x m ) notation for the elements of ( R d ) m ( x i , . . . , x di ) the (real) coordinates of x i in the above notation∆ the diagonals { x i = x j } , i = j in the above notation π W orthogonal projection onto Wπ i orthogonal projection onto the i :th coordinate π i orthogonal projection onto the [ m ] \ { i } coordinates E ( ε ) open ε -neighbourhood of a set E ⊂ R d E ◦ interior of EE closure of E Table 1.
Summary of notation P : R M → R q . Let P r,q,M denote the family of polynomials R M → R q of degree ≤ r and write P reg r,q,M for the polynomials in P r,q,M for which 0 is a regular valueon [0 , M . We identify elements P = ( P , . . . , P q ) of P r,q,M with the coefficients of P i , i ∈ [ q ] and in this way see P r,q,M as a subset of some Euclidean space. TheEuclidean distance between the coefficients of P , P ∈ P r,q,M , induces a metric on P r,q,M and this will be denoted by | P − P | .Throughout the paper, C, C ′ , C , etc., denote positive deterministic constantswhose precise value is of no importance (and their value may change from line toline), while K, K ′ , K etc. will always denote random positive real numbers.Our notation is summarized in Table 1. These will be specified later wheneverneeded. 3. Preliminaries on fractal percolation
In this section we review some standard facts about fractal percolation. It willbe convenient for us to work with fractal percolation conditioned on survival, so webegin by describing this variant.Given d ∈ N and 2 − d < p <
1, we consider fractal percolation in [0 , d withparameter p (recall the definition from the beginning of § e A n theunion of the retained cubes in Q n (the cubes that have not been removed in thefirst n generations of the fractal percolation process). Let A = A perc = A perc( d,p ) = \ n ∈ N e A n denote the fractal percolation limit set. We take the closure to ensure the compact-ness of A ; recall that the elements of Q are half-open.Since p > − d , it is well known that the limit set A is non-empty with positiveprobability. Nevertheless, for any p <
1, the probability of extinction (i.e. A = ∅ ) ispositive. We consider the surviving fractal percolation defined via the followingprocedure. Let p > − d , and given k ∈ [2 d ], denote by p k > p k = P ( There are exactly k surviving cubes Q ∈ Q | A = ∅ )where Q ∈ Q n is called surviving if for each m ≥ n , there is Q ′ ∈ Q m such that Q ′ ⊂ e A m ∩ Q ′ (expressing A via the associated dyadic tree, this means that thesub-tree rooted at the vertex corresponding to Q is infinite). Although the preciseformula is not important, we note that p k = (cid:18) d k (cid:19) p k (1 − q ) k − (1 − p (1 − q )) d − k , where q ∈ ]0 ,
1[ is the probability that A = ∅ . We recall that q is the smallest rootof f : [0 , → R , where f ( t ) = d X k =0 (cid:18) d k (cid:19) p k (1 − p ) d − k t k . ATTERNS IN RANDOM FRACTALS 13 is the probability generating function corresponding to the Galton-Watson processassociated to fractal percolation with extinction (See [25, § n ≥
0, denote by A n the union of the surviving cubes in Q n . Then, A = ∞ \ n =0 A n . and notice that with this notation, A = ∅ and A = [0 , d are the same event,so that we can condition on each of them indistinctly. We observe that the lawof A n (and whence A ) on A = ∅ is given by a Galton-Watson process with theoffspring probabilities ( p k ) k ∈ [2 d ] (see [25, Proposition 5.28] for details). In particular, A = [0 , d and for each Q ∈ Q n , conditional on Q ⊂ A n , the probability that Q ′ ⊂ A n +1 ∩ Q for exactly k cubes Q ∈ Q n +1 equals p k . Furthermore, denoting by [ A n ] the indicator function of A n and letting ν n = p − n [ A n ] , the distribution of ν n +1 | Q , Q ∈ Q n are independent conditional on B n , where B n is the sigma-algebra generated by the random sets A n . One easily checks that P ( Q ⊂ A ) = p for Q ∈ Q , and this together with the stochastic self-similarityimplies the martingale property E ( ν n +1 ( x ) | B n ) = ν n ( x ) for all x ∈ [0 , d , n ∈ N . Note that since each Q n consists of pairwise disjoint cubes, this holds also on theboundaries of the dyadic cubes. We may interpret each ν n as a measure (assigningmass ν n ( B ) = p − n L ( B ∩ A n ) to each Borel set B ⊂ R d ). It is easy to see that thissequence of measures is almost surely convergent in the weak-* sense, we denote thelimit measure by ν . The above discussion shows also that if e ν denotes the originalfractal percolation measure (defined via the retained cubes instead of the survivingones), then conditioned on e A = ∅ , the measures ν and e ν are multiples of each other.It is known (see [24, Theorem 4.1]) thatdim( ν, x ) = lim inf r ↓ log ν ( B ( x, r ))log r = s for all x ∈ A. This property implies, via the mass distribution principle, that dim H ( A ′ ) ≥ s forany set A ′ of positive ν -measure; in particular this is true for A . On the other hand,since 2 − sn |{ Q ∈ Q n : Q ⊂ A n }| is a positive martingale, we get that dim B ( A ) ≤ s and therefore dim H ( A ) = dim B ( A ) = s .Throughout the rest of the paper, we will always work with the surviving fractalpercolation as just defined, and denote the associated probability measure by P .If needed, the original definition via the sets e A n will be referred to as fractalpercolation with extinction and its law is denoted e P . To conclude, note if an event F holds P -almost surely, then e P ( F | A = ∅ ) = 1. Hence, it is enough to proveall the theorems stated in § N n = |{ Q ∈ Q n : Q ⊂ A n }| be the number of generation n cubes for (surviving) fractal percolation. Lemma 3.1.
Let C be a collection of subsets of [0 , d such that P ( C ) > (forsimplicity of notation, we denote P ( C ) = P ( A ∈ C ) and, in particular, we assumethat A ∈ C is a measurable event). Then almost surely there exists n such thatfor all n ≥ n there is a cube Q ∈ Q n such that h Q ( A ∩ Q ) ∈ C , where h Q is thehomothety renormalizing Q back to [0 , d .Proof. We claim that there is a constant δ = δ ( d, p ) > P ( N n ≤ δn ) ≤ (1 − δ ) n . (3.1)Let f ( t ) = P d k =1 p k t k be the probability generating function for the associatedGalton-Watson process. Note that f ( t ) ≤ t ( p + (1 − p ) t ) ≤ t ( p + (1 − p ) /
2) =: γ t for t ≤ /
2, so that f n (1 / ≤ γ n (note that γ < P (2 − N n > γ n ) ≤ E (2 − N n ) γ n ≤ γ n . This shows that (3.1) holds.For each n , let F n be the event that h Q ( A ∩ Q ) / ∈ C for all Q ∈ Q n making up A n . Then (3.1) gives P ( F n ) ≤ P ( N n ≤ δn ) + P ( F n | N n ≥ δn ) ≤ (1 − δ ) n + (1 − P ( C )) δn . The Borel-Cantelli lemma now yields the result. (cid:3)
As a corollary, we obtain a small variant of the standard zero-one law for Galton-Watson processes (see e.g. [25, Proposition 5.6]).
Corollary 3.2.
Let C be a collection of subsets of R d such that: (i) if E ⊂ E ′ and E ∈ C then E ′ ∈ C , (ii) any homothetic copy of E ∈ C is again in C . Furthermore,assume that the event A ∈ C is measurable.Then P ( A ∈ C ) ∈ { , } .Proof. Suppose P ( A ∈ C ) >
0. Then Lemma 3.1 ensures that almost surely h Q ( A ∩ Q ) ∈ C for some cube Q , which in light of the assumptions on C gives the claim. (cid:3) ATTERNS IN RANDOM FRACTALS 15
In our applications of these zero-one laws, C will consist of sets containing certainpatterns, such as all angles in a given open set. Another very useful basic property isHarris’ inequality (a special case of the FKG inequality). We state it in a form suitedto fractal percolation. Recall that q = q ( d, p ) denotes the extinction probability forthe fractal percolation with extinction. Lemma 3.3.
Let C , C be collections of subsets of [0 , d which are closed undertaking supersets, and such that A ∈ C i is measurable Then, P ( C ∩ C ) ≥ (1 − q ) P ( C ) P ( C ) . Proof. If P ( C ) = 1 or P ( C ) = 1 the claim is trivially true. We may thus assumethat ∅ / ∈ C ∪ C . Let e P denote the law of fractal percolation with extinction. Sincewe are assuming that ∅ / ∈ C ∪ C , it follows that(1 − q ) P ( C ) = e P ( C ) , for C = C , C , C ∩ C . (3.2)Recalling that fractal percolation with extinction corresponds to Bernoulli percola-tion on a 2 d -adic tree and C , C correspond to increasing events, Harris inequality(see [25, § e P ( C ∩ C ) ≥ e P ( C ) e P ( C ) . Combining with (3.2) gives the claim. (cid:3)
Remark . Suitable versions of Lemma 3.1 and Lemma 3.3 hold also for the finitelevel approximations A n = A perc n (with the same proofs).A classical result of Lyons asserts that for an arbitrary tree, the critical survivalpercolation parameter equals the branching number (essentially, the Hausdorff di-mension of the boundary). Representing sets via their associated dyadic trees, thisyields the the following Euclidean version; see [25, Theorem 15.11] for the proof (ofa sharper and more general result). Theorem 3.5.
Let B ⊂ [0 , d be a closed set, and let A = A perc ( d,p ) . If P ( A ∩ B = ∅ ) > , then dim H ( B ) ≥ d − s . This is very useful when B is random, because it allows to estimate the Hausdorffdimension of a random set by testing survival of a smaller random set, which is apriori an easier problem.4. A class of random measures and their intersections withparametrized families of deterministic measures
In this section we state and prove our main result on continuity of intersections.This result is presented and proved in an abstract framework. In the later sectionswe will apply this result mostly to Cartesian products of fractal percolation to deduce our geometric applications. As mentioned in the introduction, we believethat Theorem 4.9 should have similar applications to a wide variety of randommeasures including various subdivision and cut-out type random fractals. We startby defining the necessary concepts.4.1.
Random measures.
Our goal in this section is to study intersections of ran-dom measures µ with a deterministic family of measures { η t } t ∈ Γ .We consider a sequence of Borel functions µ n : R M → [0 , + ∞ ), corresponding tothe densities of absolutely continuous measures (also denoted µ n ). We note thatthese are actual functions (defined for every x ) and not equivalence classes, since wewill be integrating them against arbitrary measures. We assume that the followingstanding assumptions hold:(RM1) µ is a deterministic bounded function with bounded support.(RM2) There exists an increasing filtration of σ -algebras B n (on some space Ω) suchthat µ n is B n -measurable.(RM3) There is C < ∞ such that µ n +1 ( x ) ≤ Cµ n ( x ) for all n ∈ N , x ∈ R M .The last condition is of technical nature and could certainly be weakened. If wereplace C by a deterministic sequence C n growing at most subexponentially, thenthe proof of our main abstract theorem, Theorem 4.9, goes through with very minorchanges. The papers [3, 11] consider geometric properties of random measures whichsatisfy (RM1), (RM2), and a variant of (RM3) in which C is random and/or growsquite fast with n . This suggests that there is scope for weakening the last conditionconsiderably. Since for Cartesian products of fractal percolation, which is the focusof this article, (RM3) holds as stated, we do not consider these variants here.We now introduce the parametrized families { η t } t ∈ Γ of (deterministic) measures.We always assume the parameter space is a totally bounded metric space (Γ , d ).Our main objects of interest will be the ‘intersections’ of the random measures µ n with η t as n → ∞ , and their behaviour as t varies. Formally, we define: µ tn ( A ) = Z A µ n ( x ) dη t ( x ) , for each Borel set A ⊂ R M , n ∈ N and t ∈ Γ. We are mainly interested in theasymptotic behaviour of the total mass, and denote Y tn = k µ tn k = Z µ n ( x ) dη t ( x ) ,Y t = lim n →∞ Y tn (if the limit exists) . ATTERNS IN RANDOM FRACTALS 17
Martingale condition.
Conditions (RM1)–(RM3) by themselves are far tooweak to guarantee the convergence of µ n or the regularity of the intersections µ tn (or Y tn ). Thus, we need to impose further conditions, to at least ensure the a.s.existence of a limit measure µ . Definition . A random sequence ( µ n ) satisfying (RM1)–(RM3) will be called a martingale measure , if for all x ∈ R M and n ∈ N , E ( µ n +1 ( x ) |B n ) = µ n ( x ) . (4.1)In other words, a martingale measure is a T -martingale in the sense of Kahane[20] with the extra growth condition (RM3), and it is well known and easy to seethat, in this case, the sequence µ n converges a.s. in the weak*-sense to a randomlimit measure µ . Furthermore, for each fixed t ∈ Γ, also µ tn (and Y tn ) converges a.s.to a random limit µ t (resp. Y t = || µ t || ).4.3. Spatial independence.
Martingale measures may exhibit long range spatialdependencies; in order to obtain any results about intersections, we need to im-pose conditions that guarantee a sufficient degree of independence in the process ofdefining µ n . The best that we could hope for is that if { Q j } are dyadic cubes ofside length 2 − n that are pairwise disjoint then, conditioned on the n th step of theconstruction, the masses { µ n +1 ( Q j ) } are independent random variables. This is thecontent of the next definitions originating from [36]. Definition . A sequence ( µ n ) n ∈ N satisfying (RM1)–(RM3) is uniformly spa-tially independent (USI) if there exists a constant C > C − n )-separated family Q of dyadic cubes of side-length 2 − ( n +1) , the restrictions { µ n +1 | Q |B n } are independent. Definition . A sequence ( µ n ) n ∈ N satisfying (RM1)–(RM3) is called spatiallyindependent (with respect to the family { η t : t ∈ Γ } ) if there exists a constant C > t ∈ Γ, any n ∈ N , and for any C − n -separated family Q of dyadic cubes of side-length 2 − ( n +1) , the random variables { µ tn +1 ( Q ) |B n } Q ∈Q areindependent.The paper [36] deals with martingale measures which are spatially independent(these will be termed SI-martingales for short). In order to handle cartesianproducts of independent fractal percolations, we will need to allow some long-rangedependencies between the masses µ tn +1 ( Q ) as long as they are “sparse” with largeprobability. In order to define this notion formally, we recall the concept of depen-dency graph: given an index set I , a graph with vertex set I is a dependencygraph for a family of random variables { X i : i ∈ I } if for any i ∈ I and any subset J ⊂ I such that there is no edge from i to any element of J , the random variable X i is independent from { X j : j ∈ J } . Definition . Let ( µ n ) n ∈ N be a martingale measure, and let { η t } t ∈ Γ be a family ofmeasures. The dependency degree at step n , denoted DI n , is the smallest con-stant Ψ such that, for all t ∈ Γ, there is a dependency graph for { µ tn +1 ( Q ) |B n } Q ∈Q n +1 of degree at most Ψ.Clearly, if ( µ n ) is spatially independent, then DI n is bounded over all n . In manycases ( µ n ) will only be weakly spatially dependent, in the sense that DI n will growat a sufficiently slow rate: Definition . A sequence ( µ n ) n ∈ N satisfying (RM1)–(RM3) is weakly spatiallydependent (WSD) with parameter δ ≥ { η t : t ∈ Γ } )if there is a random sequence Ψ( n ), such that the following holds.(1) DI n ≤ Ψ( n ) for all n .(2) For each ε >
0, there exist a (deterministic) C = C ε > ε n > P n ε n < ε , and B n -measurable events C n ⊂ { Ψ( n ) ≤ C δn } such that for n ≥ P C n − ( C n |B n − ) ≥ − ε n , where we denote C = Ω (the entire probability space).Here, and in the sequel, given a positive probability event C , we denote by P C the induced conditional probability distribution, i.e. P C ( F ) = P ( F | C ). We alsoremark that P ( ·|B n ) is a random variable, and hence the WSD condition requiresthat whatever the realization of µ n (so long as C n is satisfied), there is a very largeprobability that C n +1 again is satisfied.The reason we introduce the random sequence Ψ( n ), rather than dealing with DI n directly, is that in practice we will have information about certain naturaldependency graphs, while the minimum in the definition of DI n is an awkwardquantity to work with.This definition can be motivated by the example µ n = ν (1) n × ν (2) n × ν (3) n , with ν ( i ) n independent realizations of the fractal percolation measure on [0 , µ n +1 ( I i, × I i, × I i, ) are independent, conditional on B n , provided that I i,j are all contained in different intervals in Q n . However, given I i , J ∈ Q n +1 , then(for example) µ n +1 ( J × I × I ) and µ n +1 ( I × I × J ) are certainly not independentgiven B n . We then see that one can bound the dependency degree in terms of themaximum of the cardinalities of intersections of A (1) n × A (2) n × A (3) n with planes inprincipal directions, and this will allow us to show weak spatial dependence with asuitable small δ . ATTERNS IN RANDOM FRACTALS 19
H¨older continuity of intersections.
The role of the spatial independence(or weak dependence) condition is to ensure that, with overwhelming probability,the convergence of Y tn is very fast. More precisely, we decompose Y tn +1 − Y tn = P Q ∈Q n X tQ , where X tQ = Z Q ( µ n +1 − µ n ) dη t . Weak dependence ensures that there is enough independence among the X tQ thatits sum is tightly concentrated around the mean, implying that Y tn +1 is very close to Y tn with very large probability. These ideas are made precise in Lemma 4.7, whichis a small adaptation of [36, Lemma 3.4]. Before stating this lemma, we recall thefollowing definition. Definition . We say that a measure η has Frostman exponent κ ≥
0, if thereexists a constant
C > η ( B ( x, r )) ≤ Cr κ for all x ∈ R M , < r < . (4.2)The family { η t } t ∈ Γ has Frostman exponent κ if each η t satisfies (4.2) with a uniformconstant C . Lemma 4.7.
Let ( µ n ) be a martingale measure, and let η be a measure with Frost-man exponent κ ≥ . Fix n , positive constants Ψ , Υ , and write F for the event DI n ≤ Ψ , sup x ∈ R M µ n ( x ) ≤ Υ , and suppose that (4.1) holds for x ∈ supp ( η ) .Then, for any ̺ > with ̺ κn Υ − Ψ − ≥ c > , (4.3) it holds that P F (cid:12)(cid:12)(cid:12)(cid:12)Z ( µ n +1 − µ n ) dη (cid:12)(cid:12)(cid:12)(cid:12) ≥ ̺ sZ µ n dη | B n ! ≤ O (cid:0) exp (cid:0) − Ω (cid:0) ̺ κn Υ − Ψ − (cid:1)(cid:1)(cid:1) , where the implicit constants depend only on c , the ambient dimension M , and theconstant C in the definition of Frostman exponent of η . We underline that the lemma provides a uniform bound over all realizations of µ n in the event F . Of course, this implies that same bound holds conditioningonly on F , but knowledge of this is not enough for us. In the proof we will usea generalization of Hoeffding’s inequality due to Janson [19, Theorem 2.1], whichallows for dependencies among the random variables: Lemma 4.8.
Let { X i : i ∈ I } be zero mean random variables uniformly bounded by R > , and suppose there is a dependency graph with degree Ψ . Then P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ I X i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ̺ ! ≤ (cid:18) − ̺ (Ψ + 1) | I | R (cid:19) . (4.4) Proof of Lemma 4.7.
We condition on a realization of B n which is contained in F .We will obtain the desired bound irrespective of the specific realization, hence es-tablishing the claim. By assumption, there is C > η ( Q ) ≤ C − κn forall Q ∈ Q n . Define dη n = µ n dη .We decompose Q n +1 into the families Q ℓn +1 = { Q ∈ Q n +1 : C Υ2 − κℓ < η n ( b Q ) ≤ C Υ2 κ (1 − ℓ ) } , where b Q ∈ Q n is the dyadic cube containing Q . Since η n ( Q ) ≤ Z Q Υ dη ≤ C Υ2 − κn for Q ∈ Q n , we see that Q ℓn +1 is empty for all ℓ ≤ n .For each Q ∈ Q n +1 , let X Q = η n +1 ( Q ) − η n ( Q ). Then E ( X Q ) = 0 for all Q ∈ Q n +1 ;recall that we are conditioning on B n . Also, by (RM3), | X Q | ≤ O ( η n ( Q )) ≤ O (1)2 − κℓ Υ for all Q ∈ Q ℓn +1 . Moreover, since k η n k = P Q ∈Q n η n ( Q ), we have |Q ℓn +1 | ≤ O (1)2 κℓ Υ − k η n k . Furthermore, by definition of the dependency index, there is a dependency graph for { X Q : Q ℓn +1 } of degree at most Ψ. Therefore, by the Hoeffding-Janson inequality(4.4), P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈Q ℓn +1 X Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ̺ p k η n k ℓ − n ) = O (cid:0) exp (cid:0) − Ω (cid:0) ( ℓ − n ) − ̺ κℓ Υ − Ψ − (cid:1)(cid:1)(cid:1) . for any ̺ >
0. It follows that P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z ( µ n +1 − µ n ) dη (cid:12)(cid:12)(cid:12)(cid:12) > ̺ p k η n k (cid:19) ≤ X ℓ>n P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈Q ℓn +1 X Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ̺ p k η n k ℓ − n ) = O (cid:0) exp (cid:0) − Ω (cid:0) ̺ κn Υ − Ψ − (cid:1)(cid:1)(cid:1) , for any ̺ >
0, where (4.3) is used for the last estimate. (cid:3)
The following is the main abstract result of the paper.
Theorem 4.9.
Let ( µ n ) n ∈ N be a WSDM with parameter δ , and let { η t } t ∈ Γ be afamily of measures indexed by the metric space (Γ , d ) . We assume that there arepositive constants α, κ, θ, γ such that the following holds: (H1) Γ has finite upper box-counting dimension (i.e. it can be covered by O (1) r − O (1) balls of radius r for all r ∈ (0 , ). ATTERNS IN RANDOM FRACTALS 21 (H2)
The family { η t } t ∈ Γ has Frostman exponent κ . (H3) Almost surely, µ n ( x ) ≤ αn for all n ∈ N and x ∈ R M . (H4) Almost surely, there is a (random) integer K , such that sup t,u ∈ Γ ,t = u ; n ≥ K | Y tn − Y un | θn d ( t, u ) γ < ∞ . (4.5) Further, suppose that the various parameters satisfy κ − α − δ > . (4.6) Then there is a deterministic number γ > (depending on all parameters) suchthat almost surely Y tn converges uniformly in t , exponentially fast, to a limit Y t , andthe function t Y t is H¨older continuous with exponent γ . The following lemma captures the core probabilistic argument needed in the proofof Theorem 4.9. It is only here that the weak dependence assumption gets used, viaan application of Lemma 4.7.
Proposition 4.10.
Under the assumptions of Theorem 4.9, the following holds. Let λ, B > be such that λ <
12 ( κ − α − δ ) . (4.7) For each n , let Γ n ⊂ Γ be a subset with O (exp( O ( nB ))) elements.We define Z n = max v ∈ Γ n | Y vn +1 − Y vn | ,Y n = max t ∈ Γ Y tn . Then almost surely there exists an integer K such that Z n ≤ − λn max( Y n , for all n ≥ K . (4.8) Proof.
Let Ψ( n ) be the random variable in the definition of weak martingale. Fix ε >
0, and let C ε > ε n > C n ⊂ { Ψ( n ) ≤ C ε nδ } be as in the definition ofWSDM. Denote F n = (cid:8) Z n ≤ − λn max( Y n , (cid:9) . Hence, we want to show that P (lim inf F n ) = 1.To begin, we claim that there is a deterministic n ( ε ) ∈ N such that P C n (( C n +1 ∩ F n ) c | B n ) ≤ exp(2 − cn ) + ε n +1 (4.9)for all n ≥ n ( ε ), and some constant c > n, ε .For a given v ∈ Γ n , we know from Lemma 4.7 and our assumptions that P C n ( | Y vn +1 − Y vn | > − λn p Y vn | B n ) ≤ O (cid:0) exp (cid:0) − Ω ε (cid:0) ( κ − α − δ − λ ) n (cid:1)(cid:1)(cid:1) . (4.10) Note that because the bound in Lemma 4.7 is uniform, it continues to hold afterconditioning on the B n -measurable event C n . Observe that (4.3) holds with Ω = Ω ε by (4.7). Hence, recalling that | Γ n | = O (exp( O ( nB ))), and using (4.7), P C n (cid:16) Z n > − λn Y / n | B n (cid:17) ≤ O (exp( O ( nB ))) exp (cid:0) − Ω ε (1)2 ( κ − α − δ − λ ) n (cid:1) ≤ exp(2 − cn )for c = ( κ − α − δ − λ ) /
2, provided n is large enough in terms of ε only. Since x / ≤ max( x, P C n ( F cn | B n ) ≤ exp(2 − cn ) for all n ≥ n ( ε ). As P C n ( C cn +1 | B n ) ≤ ε n +1 by assumption, the estimate (4.9) follows.Note that F k is B k +1 -measurable (but not B k -measurable). Hence it follows from(4.9) that P ( C k +1 ∩ F k | C k ∩ F k − ) ≥ − exp(2 − ck ) − ε k +1 for all k ≥ n ( ε ), and likewise if we condition only on C k . Using this and thedefinition of WSDM, for any n ≥ n ( ε ) we estimate P ∞ \ k = n F k ! ≥ P ∞ \ k = n C k ∩ F k ! ≥ P ( C n ) P ( C n +1 ∩ F n | C n ) ∞ Y k = n +1 P ( C k +1 ∩ F k | C k ∩ F k − ) ≥ P ( C ) n − Y ℓ =1 P ( C ℓ +1 | C ℓ ) P ( C n +1 ∩ F n | C n ) ∞ Y k = n +1 P ( C k +1 ∩ F k | C k ∩ F k − ) ≥ ∞ Y ℓ =1 (1 − ε ℓ ) ∞ Y k = n (cid:0) − exp(2 − ck ) − ε k +1 (cid:1) , Since P k ε k < ε , we conclude that P ( ∩ ∞ k = n F k ) > − O ( ε ) if n is sufficiently large(depending on ε ). This is what we wanted to show. (cid:3) Proof of Theorem 4.9.
Having established Proposition 4.10, the proof is a small vari-ant of that of [36, Theorem 4.1]. We give full details for the reader’s convenience.Pick constants λ, B, B such that 0 < λ < ( κ − α − δ ), 0 < B < B , and γ − θB = λB . (4.11)Also, let 0 < γ < λB . (4.12)Further, for each n , let Γ n be a (2 − nB )-dense family with O (exp( O ( nB ))) elements,whose existence is guaranteed by (H1). ATTERNS IN RANDOM FRACTALS 23
By Proposition 4.10 and (H4), almost surely there is K ∈ N such that (4.5) holdsand Z n ≤ − λn max( Y n ,
1) for all n ≥ K. (4.13)For the rest of the proof we condition on such K . Our goal is to estimate X n +1 interms of X n , where X k = sup t = u X k ( t, u ), and X k ( t, u ) = | Y tk − Y uk | d ( t, u ) γ . If d ( t, u ) ≤ − B n , we simply use (4.5) to get a deterministic bound. Otherwise, wefind t , u in Γ n such that d ( t, t ) , d ( u, u ) < − Bn and estimate | Y tn +1 − Y un +1 | ≤ I + II + III , where I = | Y tn − Y un | , II = | Y tn +1 − Y t n +1 | + | Y tn − Y t n | + | Y un +1 − Y u n +1 | + | Y un − Y u n | , III = | Y t n +1 − Y t n | + | Y u n +1 − Y u n | . The term I will be estimated inductively, for II we will use the a priori estimate(4.5) and to deal with III we appeal to Proposition 4.10.We proceed to the details. If n ≥ K and d ( t, u ) ≤ − B n then, by (4.5), | Y tn +1 − Y un +1 | ≤ ( n +1) θ d ( t, u ) γ ≤ O (1) d ( t, u ) γ − θ/B . (4.14)Hence in this regime, we get a H¨older exponent γ − θ/B = λ/B >
0, thanks to(4.11).From now on we consider the case d ( t, u ) > − B n . By definition,I ≤ X n d ( t, u ) γ . (4.15)Let t , u ∈ Γ n be (2 − Bn )-close to t, u . Using (4.5), if n ≥ K then | Y tk − Y t k | ≤ kθ − γ Bn for k = n, n + 1, and likewise for u, u , whenceII ≤ O (1)2 − ( γ B − θ − γB ) n d ( t, u ) γ . (4.16)Note that due to (4.11) and (4.12), the exponent γ B − θ − γB is positive.We are left to estimating III. We first claim thatsup n ≥ K Y n ≤ O (2 αK ) < ∞ , (4.17)recall that Y n = sup t ∈ Γ Y Vn . The point here is that O (2 αK ), although random, isindependent of n . Let n ≥ K . Using (H4) again to estimate Y tn +1 via Y t n +1 , with t ∈ Γ n , d ( t, t ) ≤ − Bn , we have Y n +1 ≤ (cid:18) max v ∈ Γ n Y vn +1 (cid:19) + O (1)2 ( θ − Bγ ) n ≤ Y n + max(1 , Y n )2 − λn + O (1)2 ( θ − Bγ ) n . Recall that we are conditioning on (4.13). Since λ > θ − Bγ <
0, and Y K = O (2 αK ), this implies (4.17).Now it follows that Z n ≤ O (2 αK )2 − λn and, in particular,III ≤ O (2 αK ) 2 − ( λ − γB ) n d ( t, u ) γ , (4.18)where λ − γB > ε > K ′ >
0, such that | Y tn +1 − Y un +1 | ≤ ( X n + K ′ − εn ) d ( t, u ) γ for all n ≥ K, t, u ∈ Γ , which immediately yields X := sup n X n < ∞ .We are left to show that almost surely Y tn converges uniformly, at exponentialspeed, since then we will have | Y t − Y u | = lim n →∞ | Y tn − Y un | ≤ Xd ( t, u ) γ . To see this, observe that (4.17), (4.8) yield that Z n decreases exponentially. Esti-mating Y tm − Y tn via Y t m − Y t n and using the estimates (4.5), (4.13), we concludethat for all t , | Y tn +1 − Y tn | ≤ | Y tn +1 − Y t n +1 | + | Y t n +1 − Y t n | + | Y t n − Y tn |≤ O ( Y K + 1)2 − λn + O (1)2 ( θ − Bγ ) n . This shows that the sequence { Y tn } is uniformly Cauchy with exponentially decreas-ing differences, finishing the proof. (cid:3) Remarks . (i) An inspection of the proof shows that one can take any γ < γ ( κ − α − δ )2( θ + λ ) . We do not expect this to be optimal (see [35] for a special case in which thesharp H¨older exponent can be determined.)(ii) The proof works with minor changes if, instead of assuming (H1), we assumethe weaker condition that Γ can be covered by exp( r − ξ ) balls of radius r forsmall r , where ξθ < γ ( κ − α − δ ) . ATTERNS IN RANDOM FRACTALS 25
Although we do not treat them here, we note that there are natural familiesthat satisfy a size bound of this kind (but have infinite box counting dimen-sion), such as Hausdorff measures on convex curves, see [37, Proposition 6.1].(iii) In all our applications, we will be able to take the random variable K of (H4)to be 1. Roughly speaking, this is because the a priori H¨older continuity willfollow from the transversality of the hyperplanes in the dyadic grid with certainalgebraic varieties, which is a purely deterministic geometric phenomenon.Allowing K to be random is useful when the martingale µ n is not tied toa fixed geometric frame, as is the case, for example, for Poissonian cutouts -see [36] for further discussion.(iv) In practice, the most important assumption on the parameters is κ > α . Oncethis holds, it is often possible to find the required 0 < δ < κ − α . In manysituations, and certainly for Cartesian powers of fractal percolation, d − α equals the dimension of the random measure µ , and thus κ > α simply meansthat dim µ + dim η t > M . When dim µ + dim η t < M , we can no longer expect Y tn to converge to a continuous (or even finite) limit. However, we will stillneed to bound the size of the intersections in this case, in the sense of havinggood control on the growth rate of Y tn . For this, we use the following variantof Theorem 4.9 and [36, Theorem 4.4]. Theorem 4.12.
Suppose that ( µ n ) is a WSDM with parameter δ satisfying (H1) – (H3) from Theorem 4.9, together with the following condition: (H6) There are γ > and deterministic families Γ n ⊂ Γ with at most exp( O ( n )) elements, such that a.s. there is a random integer K such that for each n ≥ K , sup t ∈ Γ Y tn ≤ sup t ∈ Γ n Y tn + O (2 γn ) . Suppose further that < α + δ − κ < γ . (4.19) Then, almost surely, sup n ∈ N ,t ∈ Γ − γn Y tn < ∞ . Proof.
The proof is similar to that of Proposition 4.10, and to Theorem 4.4 in [36],so we skip some details. Let Γ n ⊂ Γ be as in (H6), Z n = max v ∈ Γ n | Y vn +1 − Y vn | and Y n = max t Y tn .Firstly, we claim that almost surely there is K ∈ N such that, for some γ < γ , Z n ≤ q γ n Y n , (4.20) for all n ≥ K . Indeed, pick γ < γ so that (4.19) continues to hold with γ in placeof γ . We can apply Lemma 4.7 to get, for each fixed v ∈ Γ n , P C n ( | Y vn +1 − Y vn | > γ n/ p Y vn | B n ) ≤ exp (cid:0) − Ω ε (cid:0) ( κ + γ − α − δ ) n (cid:1)(cid:1) , where ε is as in the definition of WSDM. Recalling (4.19), we deduce that, provided n ≫ ε P C n (cid:16) Z n > γ n/ p Y vn | B n (cid:17) < exp( − cn )for c = ( κ + γ − α − δ ) >
0. From here the proof of (4.20) is concluded exactlyas in the proof of Proposition 4.10.In combination with (H6), the bound (4.20) implies that a.s. there is a randominteger K such that for all n ≥ K , Y n +1 ≤ Y n + q γ n Y n Writing K ′ = 2 − γK Y K , we conclude by induction in n ≥ K that Y n ≤ O ( K ′ )2 γn forall n ≥ K , as claimed. (cid:3) Affine intersections and linear patterns
Intersections with affine planes.
In this section we start applying Theorem4.9 to study the geometry of fractal percolation. Recall from Section 3, that A n denotes the union of the surviving cubes Q ∈ Q n . Moreover, ν n = p − n L| A n , ν = ν perc( d,p ) = lim n ν n denotes the natural measure on the (surviving) fractal percolationset A = A perc( d,p ) = spt ν = ∩ n A n . Furthermore, if p > − d , then a.s. dim H A =dim B A = s = s ( d, p ) = d + log p anddim( ν, x ) = lim inf r ↓ log µ ( B ( x, r ))log r = s for all x ∈ A (and the limit exists for ν -almost all x ). From now on, s will alwaysrefer to this number and, even when not explicitly mentioned, we assume thatthe parameters d, p have been fixed accordingly. Recall also that ν n is uniformlyspatially independent, and that ν n ( x ) ≤ αn for α = d − s = − log p and all x ∈ R d .Indeed, ν n ( x ) ∈ { , αn } for all x, n .We are interested in the Cartesian powers ( ν perc n ) m for m ≥ µ n of m independent realizations of the fractal percolation measure, which is amartingale measure. Briefly, the reason is that given distinct Q , . . . , Q m ∈ Q dn ,then conditional on Q = Q × · · · × Q m surviving, the restriction of ν m to Q has(up to rescaling and normalizing) the distribution of the product of m independent ATTERNS IN RANDOM FRACTALS 27 fractal percolations. Regarding independence, we will show that µ n is a WSDMwith parameter δ , depending on m, d, p and the parametrized family { η t } t ∈ Γ .From now on, we let ν (1) n , . . . , ν ( m ) n be independent realizations of ν perc( d,p ) n , andwrite µ n = ν (1) n × · · · × ν ( m ) n , and likewise for the limits ν ( i ) and µ . Our firstresult concerns the intersections of µ with affine planes. To formulate the result,we need a notion of angle between two affine subspaces. If V ∈ G M,k , W ∈ G M,ℓ ,we define the angle 0 ≤ ∠ ( V, W ) ≤ π/ ∠ ( V, W ) = 0 if dim( V ∩ W ) > max { , dim V + dim W − M } , and otherwise defining ∠ ( V, W ) as ∠ ( V, W ) = inf { ∠ ( v, w ) : v ∈ V , w ∈ W } where V , W are the orthogonal complements of V ∩ W inside V, W respectively.Equivalently, ∠ ( V, W ) is the j -th smallest principal angle between V and W , where j = dim( V ∩ W ) + 1. This notion of angle measures how ‘transversal’ the subspacesare. In particular, if c >
0, then the requirement ∠ ( V, W ) ≥ c can be seen as auniform transversality condition. Such conditions will arise repeatedly in the sequel. Remarks . (i) If ∠ ( V, W ) = 0 then by elementary geometry we also havesin ∠ ( V, W ) = inf x ∈ W \ V dist( x, V )dist( x, V ∩ W ) . (ii) The map ( V, W ) ∠ ( V, W ) is continuous. In particular, if dim( V ∩ W ) =max { , dim V + dim W − M } , then there exist c > V , W of V, W such that ∠ ( V ′ , W ′ ) ≥ c for all V ′ ∈ V , W ′ ∈ W . This observationwill be used repeatedly.If V ∈ A M,k and W ∈ A M,ℓ , we define the angle between V and W as theangle between the linear subspaces V ′ ∈ G M,k , W ′ ∈ G M,ℓ parallel to V and W ,respectively. For practical purposes, we also define ∠ ( V, { } ) = 1 for all V ∈ A M,k .Recall that for an index set I ( [ m ] and j ∈ [ m ] \ I , i ∈ [ d ], we denote H I = { ( x , . . . , x m ) ∈ ( R d ) m : x j = 0 for all j ∈ I } ∈ G md, ( m −| I | ) d , (5.1) H I,j,i = { x ∈ H I : x ij = 0 } ∈ G md, ( m −| I | ) d − where j ∈ [ m ] \ I, i ∈ [ d ] . (5.2) Theorem 5.2.
Let ν ( i ) n , i = 1 , . . . , m be independent realizations of ν perc ( d,p ) n , andlet µ n = ν (1) n × · · · × ν ( m ) n . Let Γ ⊂ A md,k such that for some c > and all I ( [ m ] , j ∈ [ m ] \ I , i ∈ [ d ] , each V ∈ Γ makes an angle > c with the planes H I , H I,j,i .If s = s ( d, p ) > d − k/m , then there is a deterministic γ > depending on s, k, d, m such that a.s. there is K < ∞ for which(1) The sequence Y Vn := R V µ n ( x ) d H k converges uniformly over all V ∈ Γ ;denote the limit by Y V . (2) | Y Vn − Y Wn | ≤ K d ( V, W ) γ for all n and V, W ∈ Γ , and in particular the sameholds for Y V .If s = s ( d, p ) ≤ d − k/m , then almost surely(3) sup n ∈ N ,V ∈ Γ − γn Y Vn < ∞ ,for any γ > m ( d − s ) − k .Remarks . (i) As long as Γ is compact, the transversality conditions in Theo-rem 5.2 are equivalent to assuming thatdim V ∩ H I = max { , k − m | I |} , (5.3)dim V ∩ H I,j,i = max { , k − m | I | − } , (5.4)for all I ( [ m ], j ∈ [ m ] \ I , i ∈ [ d ]. (Recall Remark 5.1 (ii).) In particular, if V ∈ A md,k satisfies (5.3), (5.4), then the transversality assumptions of Theorem5.2 are valid when Γ is a small neighbourhood of V .(ii) The transversality with respect to the coordinate hyperplanes (5.2) is needed toestablish the a priory H¨older continuity condition (H4), while the transversalitywith respect to the planes (5.1) is used to bound the dependency degree of µ n .Depending on the value of k , m , d one of these conditions may (or may not)imply the other.The theorem is a rather direct application of Theorems 4.9 and 4.12. All thehypotheses in these theorems are fairly easy to check, except for the fact that µ n is a WSDM with a suitably small parameter δ . This will be verified by a jointprobabilistic induction in n and m . See also the survey [35] for the proof of aspecial case highlighting the main ideas.Many of our arguments will feature an induction on m ; in order to set it up weneed some further notation. Given V ∈ A md,k , we let V ′ i,t = π i ( V ∩ { x ∈ ( R d ) m : x i = t } ) , ( i ∈ [ m ] , t ∈ [0 , d ) . (Recall the notation from Section 2.) For Γ ⊂ A md,k , we define R (Γ) = { V ′ i,t : V ∈ Γ , i ∈ [ m ] , t ∈ R d } . (5.5)For 2 ≤ p ≤ m −
1, we inductively define R p (Γ) = R ( R p − (Γ)). Remark . In terms of the families R (Γ), the transversality assumptions in The-orem 5.2 are equivalent to the claim that for each p such that R p (Γ) is nontrivial,all V ∈ R p (Γ) make an angle ≥ c with the planes of the type { x j = 0 } , { x ij = 0 } for all j ∈ [ m − p ], i ∈ [ d ].In the course of the proof of Theorem 5.2, we will require the following tail boundfor Y n , whose proof may be gleaned from the proofs of Theorems 4.9 and 4.12. ATTERNS IN RANDOM FRACTALS 29
Lemma 5.5.
Under the assumptions of Theorem 5.2, let δ > , λ > max { , m ( d − s ) − k + δ } , and denote Y n = sup W ∈ Γ Y Wn . Suppose Ψ( n ) is a bound for the dependency degreeas in the definition of WSDM. Let F ∈ B n such that Ψ( n ) ≤ C nδ and Y n ≤ C λn on F , where C < ∞ is a sufficiently large constant depending only on λ .Then P F (cid:0) Y n +1 ≥ C λ ( n +1) | B n (cid:1) = O (exp( − C Ω( n ) )) , with the O ( · ) , Ω( · ) are independent of C and n .Proof. In the course of the proof, the O ( · ) , Ω( · ) constants independent of n or C (but allowed to depend on any other parameters).To begin with, we verify that for each dyadic cube Q ∈ Q mdn , the map V k ( V ∩ Q ), Γ → R is Lipschitz, where the Lipschitz constant is independent of n and Q (it only depends on the transversality constant c ). To see this, fix V, W ∈ Γ,denote ε = d ( V, W ) and note that on A md,k , our metric d is equivalent to theHausdorff metric on [0 , d . It follows that π W ( V ∩ Q \ ∂Q ( O ( ε ))) ⊂ W ∩ Q , (5.6)where π W denotes the orthogonal projection onto W . On the other hand, since V forms an angle ≥ c with the faces of Q , H k ( V ∩ ( ∂Q ( O ( ε )))) = O ( ε ) . (5.7)Combining (5.6), (5.7), and using that orthogonal projection does not increase H k -measure, we get H k ( W ∩ Q ) ≥ H k ( V ∩ Q ) − O ( ε ) ≥ H k ( V ∩ Q ) − O ( d ( V, W )) . By symmetry, we end up with the estimate |H k ( W ∩ Q ) − H k ( V ∩ Q ) | = O ( d ( V, W )) , (5.8)as required. Since each V ∈ Γ intersects at most O (2 nk ) cubes in Q n , and µ n ( x ) ≤ m ( d − s ) n for all x , from (5.8) we get the (crude but sufficient) estimatesup V = W ∈ Γ | Y Vn − Y Wn | = O (cid:0) d ( V, W )2 ( m ( d − s )+ k ) n (cid:1) , (5.9)for all n ∈ N .Let Γ n +1 ⊂ Γ be Ω(2 n ( λ − m ( d − s ) − k ) ) dense. By restricting Γ to those planes thathit a fixed neighbourhood of the unit cube, we may assume that Γ is bounded. Itis then easily seen that Γ has finite upper box dimension in the metric on A md,k defined in Section 2. Thus, we may assume that Γ n +1 has O (exp( O ( n )) elements.Now (5.9) implies sup V ∈ Γ Y Vn +1 ≤ sup V ∈ Γ n +1 Y Vn +1 + 2 λn . (5.10)For each V ∈ Γ n +1 , we can use Lemma 4.7 in a similar way to (4.10) to estimate P F (cid:0) | Y Vn +1 − Y Vn | > C λn | B n (cid:1) ≤ exp (cid:0) − Ω( C )2 n ( λ + k − m ( d − s ) − δ (cid:1) ≤ exp (cid:0) − C Ω( n ) (cid:1) . Thus, if C is so large that C λ > C + C + 1, then (5.10) and the fact that Γ n +1 has O (exp( O ( n )) elements, imply the claim. (cid:3) Proof of Theorem 5.2.
It is clear that µ n is a martingale measure in R md , and that(H2), (H3) hold with m ( d − s ) in place of α , and with Frostman exponent k . Further,(H4) holds for θ = m ( d − s ) + k and γ = 1 as explained in (5.9). Also (H1) holds,and (H6) is valid for any γ >
0, since Γ has finite upper box dimension as explainedin the proof of Lemma 5.5. Thus, all the claims follow from Theorems 4.9 and 4.12if we can show that µ n is a WSDM with parameter δ (5.11)for some δ < k − m ( d − s ), if k − m ( d − s ) >
0, and for any δ >
0, if k − m ( d − s ) ≤ V + z ∈ Γ for all V ∈ Γ and all z ∈ R d for which ( V + z ) ∩ [0 , md = ∅ .Let us start by defining dependency graphs for µ Vn +1 ( Q ), Q ∈ Q mdn . We observethat if Q ∈ Q mdn and Q ′ ⊂ Q mdn are such that π j ( Q ) = π j ′ ( Q ′ ) for all j, j ′ ∈ [ m ] andall Q ′ ∈ Q ′ then, conditional on B n , µ Vn +1 ( Q ) is independent of { µ Vn +1 ( Q ′ ) } Q ′ ∈Q ′ .Hence the graph defined (for each V ∈ Γ) by drawing an edge between
Q, Q ′ ∈ Q mdn if and only if Q ∩ V = ∅ = Q ∩ V ′ and π j ( Q ) = π j ′ ( Q ′ ) for some j, j ′ ∈ [ m ], is adependency graph for { µ Vn +1 ( Q ) } Q ∈Q mdn .If k ≤ d , the transversality with respect to the hyperplanes (5.2) implies that forall j ∈ [ m ], Q ∈ Q dn , and V ∈ Γ, we havediam( V ∩ π − j ( Q )) = O c (1) . Hence, given Q ∈ Q mdn , there can be at most O c,m (1) cubes Q ′ ∈ Q mdn such that π j ( Q ) = π j ′ ( Q ′ ) for some j, j ′ ∈ [ m ]. Consequently, the dependency degree of { µ Vn +1 ( Q ) } Q ∈Q mdn is uniformly bounded, so in this case (5.11) holds for any δ > k > d and we prove (5.11) by inductionon m (with the data Ψ( n ) , C n , ε n to be specified in the course of the proof). If m = 1,the claim is clearly true since then ( µ n ) is (uniformly) spatially independent, so thatΨ( n ) = O (1), recall Definition 4.2. Suppose, then, the claim holds for m − ≥ e µ n = ν (1) n × · · · × ν ( m − n ATTERNS IN RANDOM FRACTALS 31 with the elements of R (Γ). For V ′ ∈ R (Γ), consider e Y V ′ n = Z V ′ e µ n d H k − d , and let Z n = sup V ′ ∈R (Γ) e Y V ′ n . We first claim that there is a constant C > m ) such that DI n ≤ Ψ( n ) := CZ n n ( k − d − ( m − d − s )) . (5.12)where DI n is the dependency degree of µ n with respect to Γ.Suppose that V ∈ Γ, Q ∈ Q mdn and Q ⊂ Q mdn are such that Q ⊂ A (1) n × · · · × A ( m ) n , V ∩ Q = ∅ and for all Q ′ ∈ Q it holds that Q ′ ⊂ ( A n ) m , V ∩ Q ′ = ∅ , and also π j ( Q ) = π j ′ ( Q ′ ) for some j, j ′ ∈ [ m ]. If we can show that |Q| = O m ( Z n n ( k − d − ( m − d − s )) ) , (5.13)then (5.12) follows by virtue of the dependency graphs defined above.We note that it suffices to show (5.13) in the case when j is fixed and j ′ = j forall Q ′ ∈ Q . Indeed, since there are only m possible pairs ( j, j ′ ), there is a subset Q ′ ⊂ Q with |Q ′ | ≥ |Q| /m and j, j ′ ∈ { , . . . , m } such that the above conditionsare satisfied with a fixed j, j ′ for all Q ′ ∈ Q ′ . Furthermore, replacing Q by any Q ′ ∈ Q ′ (and Q ′ by Q ′ \ { Q ′ } ), we may assume that j = j ′ . Thus, in the followingwe assume that Q, Q are as above and j = j ′ is fixed.Without loss of generality, we may assume that V contains the origin so that V ∈ G md,k and furthermore, that 0 ∈ π j ( Q ) (this is just to simplify notation). Let V = V ∩ { x j = 0 } , let V ⊥ ∈ G md,md − k be the orthogonal complement of V , and let e V = V ∩ V ⊥ ∈ G md,d the orthogonal complement of V relative to V . Then, the map π j | e V : e V → π j ( e V ) is O (1 /c )-Bi-Lipschitz, where c > x, y ∈ e V and x = y then, using Remarks 5.1(i),sin c ≤ dist( x − y, { x j = 0 } )dist( x − y, V ∩ { x j = 0 } ) = | π j ( x ) − π j ( y ) || x − y | ≤ V forms an angle ≥ c with the plane { x j = 0 } ∈ G md, ( m − d .Let us denote U = ∪ Q ∈Q Q , B = B (0 , √ md − k − n ) ⊂ V ⊥ ,B = B (0 , √ dO (1 /c )2 − n ) ⊂ e V , and A ′ n = A (1) n × · · · × A ( m ) n , e A ′ n = A (1) n × · · · × A ( m − n . Using Fubini’s theorem, we arrive at the estimate |Q| − nmd = L md ( U )= Z y ∈ B Z z ∈ B H k − d ( U ∩ ( V + y ) ∩ { x j = π j ( z ) } ) d L d ( z ) d L md − k ( y ) ≤ O (2 − n (( m +1) d − k ) ) sup y ∈ R md ,z ∈ R d H k − d ( A ′ n ∩ ( V + y ) ∩ { x j = π j ( z ) } ) , (5.15)with the O constant depending on c, d, m, k . Noting that H k − d ( A ′ n ∩ ( V + y ) ∩ { x j = π j ( z ) } ) = H k − d (cid:16) e A ′ n ∩ V ′ (cid:17) = 2 n ( m − s − d ) e Y V ′ n , for V ′ = π j (( V + y ) ∩ { x j = π j ( z ) } ) ∈ R (Γ), (5.15) yields (5.13).To complete the proof of (5.11) we still need to verify that Ψ( n ) as defined in(5.12) fulfils the conditions in the definition of WSDM.Write ξ = k − d − ( m − d − s ) for simplicity. If ξ >
0, we know by the inductionassumption (recall Remark 5.4) that e µ n is a WSDM (with respect to R (Γ)) withparameter b δ , for some b δ < ξ , while if ξ ≤
0, then this holds for any b δ >
0. Choosinga suitable λ > max { , b δ − ξ } , (5.16)and considering different cases depending on the signs of k − m ( d − s ) = ξ + s and ξ , we can make sure that for 0 < δ := λ + ξ , (5.17)we have δ < k − m ( d − s ) if s > d − k/m , while δ can be arbitrarily small if s ≤ d − k/m .Having such parameters b δ, λ, δ fixed, given ε >
0, there are
C < ∞ and events b C n ⊂ { b Ψ( n ) ≤ C b δn } such that P ( b C ) > − ε and P (cid:16)b C n +1 | B n (cid:17) ≥ − ε n +1 on b C n ,where P n ε n < ε . Here b Ψ( n ) is the bound for the dependency degree for e µ n and V ′ ∈ R (Γ) as in the definition of WSDM.Let us define E n = b E n ∩ { Z n ≤ C λn } . Thanks to (5.16), and making C largerif needed, we may apply Lemma 5.5 to e µ n and R (Γ). We deduce that P C n ( C n +1 | B n ) ≥ − ε n +1 − O (cid:0) exp( − C Ω( n ) ) (cid:1) . (5.18)Since P n ε n + O (exp( − C Ω( n ) )) can be made as small as we wish by making ε smallenough, and C large enough, and asΨ( n ) = CZ n − nξ ≤ C n ( λ − ξ ) = C δn on C n , we conclude that ( µ n ) is a WSDM with a parameter δ obeying the desiredbounds. (cid:3) ATTERNS IN RANDOM FRACTALS 33
Remark . All of our applications of Theorem 5.2 deal with the case s > d − k/m .However, in the induction step, the case s ≤ d − k/m is also needed.Theorem 5.2 gives non-trivial information only if there is a positive probabilitythat Y V > V ∈ Γ. We next show that, for a fixed V for which it is a prioripossible to have P ( Y V > >
0, this is indeed true. In the spatially independentcase, this was proved in [36, Lemma 3.6] via a tail estimate. Here we use the secondmoment method.
Lemma 5.7.
In the setting of Theorem 5.2, suppose that s > d − k/m . Then, foreach V ∈ Γ with V ∩ ]0 , md = ∅ , P ( Y V > > . Proof.
It is enough to show that Y Vn = Z V µ n d H k is an L bounded martingale, as the claim then follows from the martingale con-vergence theorem. Since Y Vn is clearly a martingale, it remains to establish L boundedness.Let Q mdn ( V ) denote the cubes in Q mdn that hit V . To begin, we estimate E ( Y n ) = 2 n ( m ( d − s )) E (cid:16)(cid:0) H k ( V ∩ A n ) (cid:1) (cid:17) = 2 n ( m ( d − s )) X Q,Q ′ ∈Q mdn ( V ) P ( Q ∪ Q ′ ⊂ A n ) H k ( V ∩ Q ) H k ( V ∩ Q ′ ) ≤ O (1)2 n ( m ( d − s ) − k ) X Q,Q ′ ∈Q mdn ( V ) P ( Q ∈ A n ) P ( Q ′ ∈ A n | Q ∈ A n )= O (1)2 n ( m ( d − s ) − k ) n ( m ( s − d )) X Q ∈Q mdn ( V ) X Q ′ ∈Q mdn ( V ) P ( Q ′ ∈ A n | Q ∈ A n )= O (1)2 n ( m ( d − s ) − k ) max Q ∈Q mdn ( V ) X Q ′ ∈Q mdn ( V ) P ( Q ′ ∈ A n | Q ∈ A n ) . (5.19)Hence, we fix Q ∈ Q mdn ( V ) and set out to estimate ζ ( Q ) := X Q ′ ∈Q mdn ( V ) P ( Q ′ ∈ A n | Q ∈ A n ) . For r = 0 , , . . . , n , let Q r be the cube in Q mdr containing Q . We will estimate each ζ r ( Q ) := X Q ′ ∈Q mdn ( V ) ,Q ′ ⊂ Q r \ Q r +1 P ( Q ′ ∈ A n | Q ∈ A n ) separately. Given ℓ ∈ { , , . . . , m } , let T r,ℓ ( Q ) = { Q ′ ∈ Q mdn ( V ) : Q ′ ⊂ Q r \ Q r +1 , |{ j ∈ [ m ] : π j ( Q ) = π j ( Q ′ ) }| = ℓ } . We claim that |T r,ℓ ( Q ) | = O (1) max (cid:0) , ( n − r )( k − ℓd ) (cid:1) . (5.20)Indeed, fix a subset I ⊂ [ m ] with | I | = ℓ . Let e π ( x ) = ( π j ( x ) : j ∈ I ), R md → R dℓ .By the transversality assumption, V makes an angle Ω(1) with e π − (0) (this can beseen from dimensional considerations). Hence, V ( √ d − n ) ∩ e π − ( e π ( Q )) ∩ Q r can becovered by a cube of side-length O (2 − n ), if k − dℓ ≤
0, and by a parallelepiped thathas k − ℓd sides of length O (2 − r ) and the remaining md − ( k − ℓd ) sides of length O (2 − n ), otherwise. Hence, if there are M cubes R ∈ Q mdn such that R ∩ V = ∅ , R ⊂ Q r and e π ( R ) = e π ( Q ) then, comparing volumes, M = O (1) if k − dℓ ≤
0, andotherwise M − nmd ≤ O (1)2 − r ( k − ℓd ) − n ( md + ℓd − k ) . Since there are finitely many maps e π to consider, we get (5.20).On the other hand, if Q ′ ∈ T r,ℓ ( Q ), then P ( Q ′ ∈ A n | Q ∈ A n ) = 2 ( s − d )( n − r )( m − ℓ ) . (5.21)Combining (5.20) and (5.21), we get ζ r ( Q ) = m − X ℓ =0 (cid:0) O (1) max (cid:0) , ( n − r )( k − ℓd ) (cid:1)(cid:1) (cid:0) ( s − d )( n − r )( m − ℓ ) (cid:1) = O (1) ⌊ k/d ⌋ X ℓ =0 ( n − r )( k − ℓd +( s − d )( m − ℓ )) + O (1) m X ℓ = ⌈ k/d ⌉ ( n − r )( s − d )( m − ℓ ) = O (1)2 ( n − r )( k +( s − d ) m ) + O (1)= O (1)2 ( n − r )( k +( s − d ) m ) , using that k + ( s − d ) m ≥ ζ ( Q ) = n X r =0 ζ r ( Q ) = O (1)2 n ( k +( s − d ) m ) . Plugging this into (5.19), we conclude that E ( Y n ) = O (1), as desired. (cid:3) As we have remarked, our ultimate goal is to study Cartesian powers of the samefractal percolation set or measure, rather than independent copies. The followingcorollary will help us in achieving this.
ATTERNS IN RANDOM FRACTALS 35
Corollary 5.8.
Under the same hypotheses of Theorem 5.2, the following holds.Let n ∈ N and U be a finite union of cubes Q ∈ Q mdn , such that U ∩ ∆ = ∅ ,where ∆ = { x ∈ ( R d ) m : x i = x j for some i = j } is the union of all the diagonals.Then the conclusions of Theorem 5.2 also hold for Y Vn = Z U ∩ V ν mn d H k . Furthermore, if s > d − k/m , V ∈ Γ and V ∩ U ◦ = ∅ , then P ( Y V > > .Recall that U ◦ denotes the interior of U Proof.
For each Q ∈ Q mdn , Q ⊂ U , either Q ∩ A mn = ∅ , or µ n | Q is (up to scalingand renormalizing) a product of independent fractal percolation measures on theprojections π i ( Q ) ⊂ R d . Hence we can condition on B n and apply Theorem 5.2 toeach of these restrictions.Similarly, the proof of Lemma 5.7 applies to µ | U , yielding that P ( Y V > > V ∩ U ◦ = ∅ . (cid:3) Finite patterns in fractal percolation.
We now apply Theorem 5.2 andCorollary 5.8 to prove the existence of homothetic copies of finite sets in the fractalpercolation limit set. Recall that S ′ ⊂ R d is a homothetic copy of S ⊂ R d if thereare λ > z ∈ R d , such that S ′ = F λ,z ( S ), where F λ,z ( x ) = λ ( x + z ). Corollary 5.9. If m ∈ N ≥ and s > d − ( d + 1) /m , then a.s., the fractal percolationlimit set A = A perc ( d,p ) contains homothetic copies of all m -point sets { a , . . . , a m } ⊂ R d .Proof. LetΓ = { T = ( t , . . . , t m − ) ∈ ( R d ) m − : 0 = t ki = t kj if i = j ∈ [ m − , k ∈ [ d ] } . and denote V T = { ( y, . . . , y ) + λ ( t , . . . , t m − ,
0) : y ∈ R d , λ ∈ R } ∈ G md,d +1 . Ourgoal is to apply Corollary 5.8 to the family { V T } T ∈ Γ , and µ n = ( ν perc( d,p ) n ) m . Fix T ∈ Γ. Then, V T has transversal intersections with all coordinate planes { x ki = 0 } , { x i = 0 } for all i ∈ [ m ] , k ∈ [ d ]. Indeed, it is clear that V T ∩ { x ki = 0 } ∈ A md,d while, denoting t m = 0, V T ∩ { x i = 0 } = { ( λ ( t − t i ) , . . . , λ ( t m − t i )) : λ ∈ R } is a line. Furthermore, if i, j ∈ [ m − j = i and k ∈ [ d ], then V T ∩ { x j = 0 , x ki = 0 } = { } (5.22)thanks to the assumption 0 = t ki = t kj for i = j . Note that these are the onlynontrivial transversality conditions to be checked since dim V T = d + 1. For each fixed T ∈ Γ, we may find n ∈ N and disjoint cubes Q , . . . , Q m ∈ Q dn such that V T intersects the interior of Q = Q × . . . × Q m ∈ Q mdn and Q ∩ V − T = ∅ ,where V − T = { ( x, . . . , x ) + λ ( T ,
0) : x ∈ R d , λ ≤ } . Applying Corollary 5.8, we conclude that Y Tn = R Q ∩ V T µ n d H d +1 converges in aneighbourhood U of T , and that T Y T = lim n Y Tn is H¨older continuous on U .Furthermore, P ( Y T > >
0, by the last claim of Corollary 5.8. Making U smallerif needed, we can then ensure that Q ∩ V − T = ∅ for all T ∈ U , and P ( Y T > T ∈ U ) > . (5.23)If Y T >
0, then V T ∩ A m ∩ Q = ∅ . In other words, there are x ∈ A ∩ Q and λ > x + λt i ∈ A for all i = 1 , . . . , m − { t , . . . , t m − , } for all t =( t , . . . , t m − ) ∈ U is invariant under homotheties and passing to supersets. Corol-lary 3.2 then implies that a.s., A contains a homothetic copy of { t , . . . , t m − , } forall ( t , . . . , t m − ) ∈ U . Since Γ is σ -compact, we may cover Γ by countably manysuch neighbourhoods U , and conclude that a.s., A contains a homothetic copy ofany m -element set { a , . . . , a m } ⊂ R d with a ki = a kj for all i = j and k ∈ [ d ].To show that A contains also homothetic copies of those { a , . . . , a m } with a ki = a kj for some i = j , k ∈ [ d ], we have to modify the argument slightly. This is dueto the fact that if there are coincidences t ki = t kj for some i = j , k ∈ [ d ], or ifsome t ki = 0, then R ( { V T } ) contains lines parallel to some hyperplane { x ji = 0 } ∈ A ( m − d, ( m − d − .Note that in the proof of Theorem 5.2, the transversality with respect to the co-ordinate hyperplanes is only used to obtain the a priori H¨older bound via (5.8). Thekey to solving this problem is the observation that in order for the proof of Theorem5.2 to work, we may freely use any metric on R (Γ) as long as R (Γ) has finite boxdimension with respect to this metric and (5.8) remains valid. Furthermore, insteadof the whole of R (Γ), we may consider each R i (Γ) := { V ′ i,t : V ∈ Γ , t ∈ [0 , d } separately, and even use a different metric on each R i (Γ), i ∈ [ m ]. Note that inTheorem 5.2, R (Γ) = ∪ i ∈ [ m ] R i (Γ), but in the proof, the induction assumption on e Y V ′ n is applied to one R i (Γ) at a time.Now to the details. We consider different coincidence classes separately: let N ⊂ { ( i, j, k ) : i, j ∈ [ m ] , k ∈ [ d ] , i = j } ATTERNS IN RANDOM FRACTALS 37 and (still denoting t m = 0), defineΓ N = { T = ( t , . . . , t m − ) ∈ ( R d ) m − : t ki = t kj if ( i, j, k ) ∈ N and t ki = t kj if i = j and ( i, j, k ) / ∈ N } . Fix N and i ∈ [ m ]. Given T ∈ Γ N , and V = V T , the projections V ′ i,c = π i ( { V T ∩ { x i = c } )are parallel to H i = π i (cid:0) { x ∈ R d ( m − : x kj = 0 whenever ( i, j, k ) ∈ N } (cid:1) and transversal with respect to the coordinate hyperplanes orthogonal to H i .To define a new metric on R i (Γ N ), let d denote the dyadic metric on H ⊥ i . For x, y ∈ H ⊥ i , this is defined as the side length of the largest dyadic cube in Q dim H ⊥ i that contains both x and y (or 0 if x = y ). Note that d is indeed a metric since thefamilies Q n consist of pairwise disjoint dyadic cubes. The argument now carries onby defining a metric on A d ( m − , as d ( ℓ, ℓ ′ ) = max { d ( e π ( ℓ ) , e π ( ℓ ′ )) , d ( e π ( ℓ ) , e π ( ℓ ′ )) } , where e π denotes the orthogonal projection onto H i , e π is the orthogonal projectiononto H ⊥ i , and d denotes our usual metric on A dim H i , .Note that since ℓ, ℓ ′ ∈ Γ N , the projections e π ( ℓ ) , e π ( ℓ ′ ) are points, and so d iswell defined, and it is easy to check that it is indeed a metric. Furthermore, foreach fixed T ∈ Γ N , and for all ℓ ∈ R i ( { V T } ), the projections e π ( ℓ ) are transversalwith respect to the coordinate hyperplanes in e π ([0 , ( m − d ) ⊂ H i . This allowsus to conclude that for a small neighbourhood U of T in Γ N , the maps ℓ ( Q ∩ ℓ ), R ( U ) → [0 , + ∞ [ are Lipschitz continuous with respect to the d -metric,with a Lipschitz constant independent of Q ∈ Q ( m − d . Indeed, using the expression ℓ = e π ( ℓ ) × e π ( ℓ ), the Lipschitz continuity is easy to check with respect to bothcoordinates; for e π it is the same argument as in (5.9) and for e π it follows from thedefinition of the dyadic metric. Note also that (Γ N , d ) has finite box dimension.We conclude that the claims of Lemma 5.7 and Corollary 5.8 continue to hold in U (note that transversality with respect to the planes (5.2) is not used at all inthe proof of Lemma 5.7). This yields P ( Y T > T ∈ U ) > { t , . . . , t m − , } for all t ∈ Γ N by virtue of Corollary 3.2. Since there are only finitely many coincidence classes N ,this finishes the proof. (cid:3) Remark . The special case m = 2 implies that the fractal percolation set con-tains all directions almost surely, provided s > ( d − /
2. A small variant of theproof of Theorem 1.1 yields the following generalization: if s > ( d − k ) /
2, then a.s.for all k -planes V , there are two points in A perc( d,p ) determining a direction in V . We turn to finding translated copies of finite sets inside the fractal percolationset A (without scaling). Just because A is bounded, we cannot hope to find atranslation of every m point set. However, if s > d − d/m , we have the followingvariant of Corollary 5.9 Corollary 5.11.
Let X be a compact subset of ([0 , d ) m \ ∆ , where ∆ = { x i = x j for some i = j } is the union of the diagonals. If s > d − d/m , then with positiveprobability A = A perc ( d,p ) contains a translated copy of each ( x , . . . , x m ) ∈ X .Proof. The proof is very similar to the first part of the proof of Corollary 5.9 above.Fix T = ( x , . . . , x m − ) ∈ ( R d ) m − . We first find n ∈ N and a finite union U ofcubes Q ∈ Q mdn which do not intersect the ∆, and such that V T ∩ U ◦ = ∅ , wherefor T = ( t , . . . , t m − ) ∈ ( R d ) m − , V T = { ( x, . . . , x ) + ( t , . . . , t m − ,
0) : x ∈ R d } ∈ A md,d . We apply Corollary 5.8 in a suitably small neighbourhood U of T to deduce thatthere is a positive probability that Y T > T ∈ U , where Y T = lim n Y Tn and Y Tn = Z U µ n d H d | V T . The application of Corollary 5.8 is justified, since each V T is transversal to thecoordinate planes { x ji = 0 } , { x i = 0 } ( i ∈ [ m ], j ∈ [ d ]). Recall that these are theonly transversality conditions to be checked since dim V T = d . If Y T >
0, then V T ∩ A m = ∅ and so there is x ∈ A such that x + { t , . . . , t m − , } ⊂ A .Since X is compact, we can cover it by finitely many neighbourhoods U as above.The claim then follows from Harris’ inequality (Lemma 3.3). (cid:3) Optimality of the results.
As noted in the introduction, there are sets A ⊂ R d of zero Hausdorff dimension which contain homothetic copies of all finitesets of R d . This suggests that other notions of dimension might be more naturalfor this problem. If we instead consider packing dimension, we find that the dimen-sion thresholds in Corollaries 5.9, 5.11 are optimal (up to the endpoint), even fordeterministic sets: Proposition 5.12.
Let A ⊂ R d be a Borel set of packing dimension s .(1) If A contains a homothetic copy of { t , . . . , t m − , } for a non-empty openset of T = ( t , . . . , t m − ) ∈ ( R d ) m − , then s ≥ d − ( d + 1) /m .(2) If A contains a translated copy of { t , . . . , t m − , } for a non-empty open setof T ∈ ( R d ) m − , then s ≥ d − d/m .Moreover, in both cases ‘non-empty open set’ may be replaced by ‘set of full packingdimension’ in the relevant ambient space. ATTERNS IN RANDOM FRACTALS 39
Proof.
The only properties of packing dimension that we use are (a) it does notincrease under locally Lipschitz maps, and (b) dim P ( A q ) ≤ q dim P ( A ) for q ∈ N , A ⊂ R d . For the latter property see e.g. [28, Theorem 8.10].Let g be the map( x , x , . . . , x m )
7→ | x − x | − ( x − x , . . . , x m − x ) : ( R d ) m → S d − × ( R d ) m − , which is locally Lipschitz outside of ∆. Hence d − m − d = dim P ( g ( A m \ ∆)) ≤ m dim P ( A )giving (1). The claim (2) obtained in a similar manner, by considering the map( x , x , . . . , x m ) ( x − x , . . . , x m − x ) : ( R d ) m → ( R d ) m − . The last claim is clear from the argument. (cid:3)
What happens at the threshold? We recall that fractal percolation with thecritical parameter p = 2 − d goes extinct a.s. Much more is true: if V ⊂ [0 , d isa Borel set with finite ( d − s )-capacity, and A = A perc( d,p ) has dimension s , then A ∩ V = ∅ almost surely, see [25]. The usual proofs of these facts do not seemto easily extend to the setting of the product of independent realizations of fractalpercolation. Nevertheless, we have the following result: Proposition 5.13.
Let A (1) , . . . , A ( m ) be independent realizations of A perc ( d,p ) . Thenfor each compact set V ⊂ ( R d ) m of finite ( d − s ) m -dimensional Hausdorff measure,almost surely V ∩ ( A (1) × · · · × A ( m ) ) = ∅ .Proof. Without loss of generality, V ⊂ [0 , md . By assumption, there exists aconstant C (depending only on H ( d − s ) m ( V )) and collections C n of dyadic cubes suchthat:(1) C = { [0 , d } .(2) The cubes in each C n are disjoint, and their union covers V ,(3) Each element of C n +1 is strictly contained in some element of C n ,(4) P Q ∈C n − ℓ ( Q )( d − s ) m ≤ C ,where ℓ ( Q ) is the side-length of Q . As usual, write A n = A (1) n × · · · × A ( m ) n , and let K n be the number of cubes Q in C n such that Q ⊂ A ℓ ( Q ) .Let F n be the event ( K n ≥ F n are decreasing by (3),and ∩ n F n is the event V ∩ ( A (1) × · · · × A ( m ) ) = ∅ . Assume that P ( ∩ n F n ) = q > F n are nested, P ( F n ) = P ( F n | F n − ) · · · P ( F | F ) . So if we can show that P ( F cn +1 | F n ) ≥ c for some constant c > n ,we are done. Note that 2 − ℓ ( Q )( d − s ) m is the probability that Q ∈ A ℓ ( Q ) . Hence, by(4), E ( K n ) ≤ C for all n . By Markov’s inequality, P ( K n > M ) ≤ C/M . Hence, if M is large enough (depending only on C, q ) then P (1 ≤ K n ≤ M ) ≥ q/
2. Write C n for the event (1 ≤ K n ≤ M ) ⊂ F n . Note that P ( F cn +1 | C n ) ≥ (1 − ( s − d ) m ) d M =: c > . (5.24)Indeed, suppose C n holds and let Q , . . . , Q M be the cubes in C n such that Q i ⊂ A ℓ ( Q i ) (so that 1 ≤ M ≤ M ). For each j = 0 , . . . , M −
1, the probability that Q j +1 ∩ A ℓ ( Q j +1 )+1 = ∅ given that Q i ∩ A ℓ ( Q i )+1 = ∅ for i = 1 , . . . , j is boundedbelow by (1 − ( s − d ) m ) d . Indeed, this number is the unconditional probability that Q j +1 has no offspring in A ℓ ( Q j +1 )+1 , and the information that the previous cubes hadno offspring can only increase the probability (if there is overlap in some coordinateprojection). Hence the probability that all Q j have no offspring is at least c , butthanks to (3) this is a sub-event of F cn +1 , giving (5.24).To conclude, note that P ( F cn +1 | F n ) ≥ P ( F cn +1 ∩ E n ) P ( F n ) ≥ q P ( F cn +1 | C n ) ≥ qc c > . (cid:3) As an immediate corollary, we get:
Corollary 5.14.
Let ( S j ) j be a countable collection of m -element sets in R d . If s = s ( d, p ) ≤ d − ( d + 1) /m , then a.s. A perc ( d,p ) does not contain a homothetic copyof any of the S j . Likewise, if s ≤ d − d/m , then a.s. A perc ( d,p ) does not contain atranslated copy of any of the S j .Proof. It is enough to show the claim for a single set S . Moreover, by the usualdecomposition of A m \ ∆ into dyadic cubes, we may replace A perc( d,p ) by the prod-uct of m independent realizations of A perc( d,p ) . The claim is then immediate fromProposition 5.13. (cid:3) We also have the following corollary of (the proof of) Proposition 5.13 that willbe required later.
Corollary 5.15.
Suppose V ⊂ R md can be covered by C m ( d − s ) n cubes in Q n forall n . Let A (1) , . . . , A ( m ) be independent realizations of A perc ( d,p ) . Then there existsa sequence q n ↓ , depending only on C , such that P ( V ∩ A (1) n × · · · × A ( m ) n = ∅ ) ≤ q n for all n ∈ N . Proof.
Let C n be the cubes in Q n that hit V , so that (1)-(4) in the proof of Proposi-tion 5.13 hold by assumption. Let C n denote the event that V ∩ A (1) n ×· · ·× A ( m ) n = ∅ .Fix q > q ≤ P ( C n ). Then the proof of Proposition 5.13 shows thatthere is c = c ( q, C ) ∈ (0 ,
1) such that q ≤ P ( C n ) ≤ (1 − c ) n . ATTERNS IN RANDOM FRACTALS 41
This is absurd if n is large enough depending on q, C , so P ( C n ) ≤ q for n sufficientlylarge depending on q, C , which yields the claim. (cid:3) Nonlinear intersections and applications
In this section, we prove a non-linear generalization of Theorem 5.2 : we willreplace the family Γ ⊂ A md,k in Theorem 5.2 by a family of k -dimensional algebraicsurfaces of bounded degree. As applications, we obtain the sharp dimension thresh-olds for the existence of many non-linear configurations in A , such as all angles,similar copies of all triangles and all small enough distances.6.1. Preliminaries.
Before stating the main result of the section, we recall somenotation and prove some preliminary results. For
M, q ∈ N fixed, q ≤ M , let P r,q,M be the family of non-constant polynomials R M → R q of degree ≤ r (as in Section5, in our applications M will be of the form M = md for some m ∈ N ). We willoften shorten this to P r when the parameters q and M are clear from context. Alsowrite P reg r,q,M for the polynomials in P r,q,M for which 0 is a regular value on [0 , M (that is, the rank of DP ( x ) is q for all x ∈ P − (0) ∩ [0 , M ). We identify elements P = ( P , . . . , P q ) of P r with the coefficients of P i , i ∈ [ q ] and in this way see P r as asubset of some Euclidean space R N . This induces a metric, given by the Euclideandistance of the coefficients, which will be referred to as the Euclidean distance on P r .Given a polynomial P : R M → R q , we denote P − (0) ∩ [0 , M by either Z P or Z ( P ). The dimensions M, q will always be clear from context. Furthermore,we denote η P = H M − q | Z ( P ) , with the convention that η P is the trivial measure if H M − q ( Z ( P )) = 0.Globally, the Euclidean metric does not match the geometric closeness of thevarieties: two polynomials P , P with | P − P | small may have completely differentzero sets Z P , Z P . The next lemma asserts that, near a polynomial for which 0 isa regular value, this does not happen. For M = 2 and q = 1, the statement of thelemma is contained in Lemma 8.8 of [36]. Lemma 6.1.
Let P ∈ P reg r,q,M such that P − (0) ∩ ]0 , M = ∅ .Then there exist neighbourhoods O of P (in P r with the Euclidean metric) and U of Z P , and a real analytic map G : O × O × U → R M such that for all P , P ∈ O :(1) Z P ⊂ U , and G ( P , P , · ) is the identity on P − (0) ∩ U ,(2) G ( P , P , · ) | P − (0) ∩ U is a diffeomorphism onto its image,and Z P ⊂ G ( P , P , P − (0) ∩ U ) ,(3) P ( G ( P , P , u )) = 0 whenever u ∈ Z P .Proof. The idea of the proof is to define G as follows: given P , P ∈ P r,q,M closeto P and x close to Z P , we define G ( P , P , x ) as the ‘first’ point in { P = P ( x ) } that is reached by moving from x orthogonally to the variety { P = P ( x ) } . Theimplicit function theorem will ensure that this function is indeed well defined andhas the claimed properties.As above, we identify P r,q,M with the appropriate Euclidean space R N . We letΦ , Φ : R N × R N × R M × R M × R q → R M × R M be given byΦ ( P , P , x, y, λ ) = (cid:0) P ( y ) − P ( x ) , y − x − DP ( y ) t · λ (cid:1) , Φ ( P , P , x, y, λ ) = (cid:0) P ( y ) − P ( x ) , y − x − DP ( x ) t · λ (cid:1) . We write ∂ Φ i /∂ ( y, λ ) for the partial derivatives of Φ i with respect to the M + q variables ( y, λ ). Then we have the following block structures for i = 1 , ∂ Φ i ∂ ( y, λ ) ( P, P, x, x,
0) = (cid:18) DP ( x ) 0 q × q I d × d − DP ( x ) t (cid:19) . Since P ∈ P reg r,q,M , for x ∈ Z P we have:det (cid:18) DP ( x ) 0 q × q I d × d − DP ( x ) t (cid:19) = det (cid:0) − DP ( x ) · DP ( x ) t (cid:1) = 0 . The implicit function theorem together with compactness then provides neighbour-hoods e O , e U of P , Z P , and real-analytic functions ( y , λ ) , ( y , λ ) : e O × e O × e U → R M × R q , such thatΦ i ( P , P , x, y i ( P , P , x ) , λ i ( P , P , x )) ≡ , i = 1 , . Let O , U be neighbourhoods of P, Z P which are compactly contained in e O , e U re-spectively.We define G := y . Note that Φ ( P , P , x, x,
0) = 0 whenever P ∈ e O , x ∈ e U ,so the uniqueness of the implicit function (assuming e O is connected as we may)ensures that G ( P , P , x ) = x . Using this, compactness, and the continuity of Φ i ,by making O , U smaller, we may ensure that G ( P , P , x ) ∈ e U whenever P , P ∈ O and z ∈ P − (0) ∩ U . Again by compactness, and making O smaller if needed, wemay guarantee that Z e P ⊂ U for all e P ∈ O . In particular, (1) holds.We claim that, perhaps after making O even smaller, if f = y ( P , P , · ) and f = y ( P , P , · ), then f f ( x ) = x for all P , P ∈ O and all x ∈ P − (0) ∩ U . Indeed,note first that f f ( x ) ∈ P − (0) is well defined. Suppose z = f f ( x ) − x = 0. Given e P ∈ O , y ∈ U , let H e P ,y be the span of the gradients of the coordinate functions of e P evaluated at y . Since H e P ,y is perpendicular to the tangent of { e P = e P ( y ) } at y ,and the map ( e P , y ) → H e P ,y is continuous at points ( e P , y ) such that y is a regularpoint of e P , we find that (making O smaller again) the angle between H P ,f ( x ) and ATTERNS IN RANDOM FRACTALS 43 z is non-zero. But it follows from the definitions that z has the form z = DP ( f ( x )) t · λ for some λ ∈ R q \ { } , which means precisely that z ∈ H P ,f ( x ) , contradicting ourhypothesis. So z = 0 as we had claimed. Taking stock, we have shown that f isthe inverse function to f , so that f : P − (0) ∩ U → R M is a diffeomorphism ontoits image.By definition of G , we have that P ( G ( P , P , x )) = 0 whenever P , P ∈ O and x ∈ P − (0) ∩ U , in particular giving (3). By making O smaller one more time, wemay assume that Z ( P ) ⊂ G ( P , P , P − (0) ∩ U ), showing that (2) holds. (cid:3) Next, we present a useful consequence of the coarea formula for submanifolds of R M . Recall that the ℓ -Jacobian J ℓ L of a linear map L : R M → R ℓ , d ≥ ℓ , is given bythe product of its singular values, which are the square roots of the eigenvalues of LL t (so that ( J ℓ L ) = det( LL t )). We denote the tangent of a submanifold S ⊂ R M at x ∈ S by T x ( S ). Proposition 6.2.
Let P ∈ P reg r,q,M , let V ⊂ [0 , M be a Borel set, and let f : U → R ℓ be a C map, where U is a neighbourhood of Z P ∩ V and ℓ ∈ [ M ] . Assume that: • inf x ∈ V ∩Z P J ℓ ( Df ( x )) ≥ c > , • For all x ∈ Z P ∩ V , the angle between the tangent plane to P − (0) at x andker ( Df ( x )) is at least c > .If, furthermore, M − q ≥ ℓ , then H M − q ( Z P ∩ V ) ≤ O ( c − c − ( M − q )2 ) Z f ( Z P ∩ V ) H M − q − ℓ ( Z P ∩ V ∩ f − ( y )) d H ℓ ( y ) . Proof.
By a standard approximation, we may assume V is open. Let S = Z P ∩ V ;it is a regularly embedded submanifold of R M . Write J Sℓ f ( x ) for the ℓ -Jacobianrelative to S ; see [22, Definition 5.3.3]. We claim that J Sℓ f ( x ) = Ω( c c ℓ ) for all x ∈ S. (6.1)Assuming this, the claim follows immediately from the coarea formula for subman-ifolds, see [22, Theorem 5.3.9], so the task is to establish (6.1).Write L = Df ( x ) by simplicity. By the singular value decomposition, afterorthonormal changes of bases in R M and R ℓ we may assume that L has the form (cid:0) D O M − ℓ × ℓ (cid:1) , (6.2)where D is an ℓ × ℓ diagonal matrix with the singular values of L on the diagonal, sothat J ℓ ( L ) = det( D ). Now let e , . . . , e ℓ be an orthonormal basis for the orthogonalcomplement W of T x ( S ) ∩ ker( L ) inside T x ( S ); by assumption, any non-zero elementof W makes an angle ≥ c with ker( L ). Let π denote orthogonal projection onto the orthogonal complement of ker( L ). We note that π − : π ( W ) → W is well definedand O ( c )-Lipschitz; this is the same argument as in (5.14). In particular, restricting π − to the cube spanned by π ( e i ), we see that the parallelogram spanned by π ( e i )has ℓ -area Ω( c ℓ ). On the other hand, from (6.2) we see that if E is the matrix with e i as columns and E ′ is the matrix with π ( e i ) as columns, thendet( LE ) = det( D ) det( E ′ ) ≥ Ω( c c ℓ ) , so that the definition of relative Jacobian gives (6.1). (cid:3) Continuity for intersections with algebraic varieties.
We are ready toextend Theorem 5.2 to intersections with algebraic varieties. We note that for d > m = 1). Recallthe notations H I , H I,i,k from Section 2.
Theorem 6.3.
Let ν ( j ) n , j ∈ [ m ] , be m independent realizations of ν perc ( d,p ) n , and let µ n = ν (1) n × · · · × ν ( m ) n . Let q ∈ [ md − and P ∈ P reg r,q,md , such that P − (0) ∩ ]0 , md = ∅ . Suppose that foreach index set I ( [ m ] , and each i ∈ [ m ] \ I , k ∈ [ d ] , the tangent planes of P − (0) at each a ∈ P − (0) ∩ [0 , md form an angle > with the planes H I , H I,i,k .Then there is a neighbourhood O of P such that:(1) Assume s = s ( d, p ) > q/m . Denoting Y Pn = Z P − (0) ∩ [0 , md µ n d H md − q , the sequence Y Pn converges uniformly over all P ∈ O and Y P = lim n →∞ Y Pn satisfies | Y P − Y P | ≤ K | P − P | γ , where γ > is a deterministic constant depending on s, r, q, d, m , and K isa finite random variable.(2) Now suppose s ≤ q/m . Then almost surely sup n ∈ N ,P ∈O − γn Y Pn < ∞ , for any γ > q − ms . We emphasize that the neighbourhood O is independent of s .The proof of the theorem depends on several lemmas that we present first. Lemma 6.4.
Let P ∈ P reg r,q,M . Then there are a neighbourhood O of P and C = C ( P ) > such that sup P ∈O sup x ∈ R M sup r> η P ( B ( x, r )) ≤ C r M − q . ATTERNS IN RANDOM FRACTALS 45
Proof.
We can find finitely many open sets U i ⊂ R M whose union covers Z P , andcoordinate projections e π i : U i → R M − q , such that e π i | U i is injective and, moreover,the angle between ker( e π i ) and the tangent plane T x ( P − (0)) is at least c > i and x ∈ P − (0) ∩ U i . Proposition 6.2 applied to P − (0) ∩ U i ∩ B ( x, r ) and themaps e π i now yields, for any x ∈ Z P and r > η P ( B ( x, r )) ≤ O c (1) X i H M − q ( e π i ( B ( x, r ))) = O c ( r M − q ) . Letting G = G ( P , P, · ) be as in Lemma 6.1 and using that B ( x, r ) ∩ Z P ⊂ G ( B ( G − ( x ) , r )), the argument extends to a sufficiently small neighbourhood of P . (cid:3) Lemma 6.5.
Let P ∈ P reg r,q,M , q ∈ [ M − . Assume that the tangent planes of Z P make an angle > with all coordinate hyperplanes. Then there are a neighbourhood O of P and C = C ( P ) > such that sup P ∈O sup H η P ( H ( δ )) ≤ C δ, where H runs over all coordinate hyperplanes, and H ( δ ) denotes the δ -neighbourhoodof H .Proof. Firstly, we claim that there is a neighbourhood O of P such thatsup P ∈O sup H H M − q − ( H ∩ Z P ) =: C ′ = C ′ ( P ) < ∞ , (6.3)where again H runs over coordinate hyperplanes. Indeed, fix t and i ∈ [ M ] and let H t,i = { ( x , . . . , x M ) : x i = t } be a coordinate hyperplane. Consider the polynomial P t,i = ( P , x i − t ) : R M → R q +1 . By the assumption that the tangents to Z P makea positive angle with H t,i , 0 is a regular value of P t,i on [0 , M . By Lemma 6.4,there are neighbourhoods O of P and U of t such that the claim holds when takingsupremum over all planes of the form { x i = u } , u ∈ U . By compactness, we concludethat (6.3) holds.Fix i ∈ [ M ], and let e π ( x ) = x i . By compactness, and making O smaller if needed,we may ensure that all tangent planes to Z P , P ∈ O make a uniformly positive anglewith all coordinate hyperplanes. Proposition 6.2 now yields η P ( H t,i ( δ )) ≤ O (1) Z t + δs = t − δ H M − q − ( e π − ( s ) ∩ Z P ) ds = O ( δ ) , using (6.3). This is what we wanted to prove. (cid:3) Proof of Theorem 6.3.
The proof proceeds along the lines of the proof of Theorem5.2. We highlight the main differences. As before, conditions (RM1)–(RM3) areclearly valid. Our parameter space will consist of a suitable neighbourhood O of P in P reg r,q,md , where the conclusions of Lemma 6.1 hold, and such that the tangentplanes of each P ∈ O at each x ∈ Z P satisfy the required transversality conditionwith respect to the planes (5.1)– (5.2). Furthermore, denoting the family of allsuch tangent planes by Γ ⊂ A md,md − q , we assume that also R p (Γ) satisfy a similarcondition. Note that due to our assumptions and Lemma 6.1, all these conditionshold in a sufficiently small neighbourhood O of P (recall Remark 5.4). In thesequel, we will apply various estimates that all hold in a small neighbourhood of P ,and we assume that O is fixed and sufficiently small such that all of these are validsimultaneously. We stress that all the induced constants are allowed to depend on P .We will show that the assumptions of Theorem 4.9 hold for µ n and the family ofmeasures η P , P ∈ O (recall that η P = H md − q | Z ( P ) ).Trivially, (H1) holds. Furthermore, (H3) holds with α = m ( d − s ). The Frostmancondition (H2) holds, with κ = md − q , thanks to Lemma 6.4.To prove (H4), we first show that P
7→ H md − q ( Q ∩ P − (0)) is uniformly Lipschitzfor all Q ∈ Q md , P ∈ O . To that end, fix P , P ∈ O and let ε = | P − P | . Write G = G ( P , P , · ), and note that G is O ( ε )-close to the identity in the C topologyby Lemma 6.1. In particular, J Z ( P ) md − q ( G )( x ) ∈ (1 − O ( ε ) , O ( ε )) for all x ∈ Z ( P ) , where J Z ( P ) md − q ( G ) is the relative Jacobian, see [22, Definition 5.3.3]. Using this and theestimate H md − q ( Z ( P )) = O (1), the area formula for submanifolds ([22, Theorem5.3.7]) applied to the submanifold S = Q ◦ ∩ Z P and the map G gives H md − q ( S ) ≤ H md − q ( G ( S )) + O ( ε ) . By Lemma 6.1 and using again that G is C close to the identity, we know that H md − q ( G ( S )) ≤ H md − q (cid:0) P − (0) ∩ Q ( O ( ε )) (cid:1) , recall Q ( δ ) is the δ neighbourhood of Q . The last two displayed equations togetherwith Lemma 6.5 yield H md − q ( Z P ∩ Q ) ≤ H md − q ( Z P ∩ Q ) + O ( ε ) , which by symmetry implies that P
7→ H md − q ( Q ∩ P − (0)) is Lipschitz with theconstant depending only on P , as claimed. Note that we are using a quite coarseestimate here since we are not taking into account the size of Q (apart from the factthat it is contained in [0 , md ).Note that each Z ( P ), P ∈ O intersects at most O (2 n ( md − q ) ) cubes in Q n . Indeed,this is true for P − ∩ U (where U is the neighbourhood from 6.1), since this isa bounded piece of a ( md − q )-dimensional embedded manifold. The claim forarbitrary P ∈ O follows from Lemma 6.1, since the image of a cube Q ∈ Q mdn ATTERNS IN RANDOM FRACTALS 47 hitting Z P under G ( P, P , · ) can be covered by O (1) cubes in Q mdn . Using this, weconclude that sup P ,P ∈O | Y P n − Y P n | = O ( | P − P | n ( m ( d − s )+ md − q ) ) , that is (H4) holds with the parameters θ = m (2 d − s ) − q and γ = 1.We note that Lemma 5.5 also holds in this setting with essentially the same proof.The only difference is in the proof of (5.9), which we have just explained.As in the proof of Theorem 5.2, the claims follow from Theorems 4.9 and 4.12 ifwe can show that µ n is a WSDM with parameter δ (6.4)for some δ < ms − q , if s > q/m , and for any δ >
0, if s ≤ q/m .Suppose first that md − q ≤ d . In this case, DI n = O (1) (deterministically),recall that DI n denote the dependency degree of µ n . To see this, note that becausethe tangent planes to Z P make an angle ≥ c > H z,j = π − j ( z ), forany P ∈ O , x = x ′ ∈ Z P ∩ H z,j we have | x − x ′ | = Ω P ,c (1). Using transversalityagain, this implies that Z P ∩ π − j ( B ( z, − n )) can be covered by O P ,c (1) balls ofradius O P ,c (2 − n ), which gives the claim.Therefore, we assume from now on that md − q > d . Again, we proceed byinduction on m . If m = 1, the claim holds since then ( µ n ) is even an SI-martingale.Suppose the claim holds for m − ≥
1. We will need an algebraic variant of R (Γ)(Recall that Γ denotes the collection of all the tangent planes of Z P , P ∈ O ). Given P ∈ O , c ∈ [0 , d , j ∈ [ m ] and y ∈ B (0 , δ ) ⊂ R q , let e P P,c,j,y = P ( x , . . . , x j − , c, x j , . . . , x m − ) − y, and set e O = { e P P,c,j,y : P ∈ O , c ∈ [0 , d , j ∈ [ m ] , y ∈ B (0 , δ ) ⊂ R q } . (6.5)Here δ is chosen small enough that the transversality assumptions continue to holdfor e P ∈ e O (this is possible thanks to the transversality assumptions for R (Γ)).For each e P ∈ e O , denote e Y e Pn = Z Z ( e P ) ν (1) n × · · · × ν ( m − n d H ( m − d − q . As in the proof of Theorem 5.2, letting Z n = sup e P ∈ e O e Y e Pn , we will show that for n ≥ O (1) there is a constant C < ∞ such thatΨ( n ) = CZ n n ( − q +( m − s ) . (6.6)is an upper bound for the dependency degree of µ n . To verify this, it suffices toshow that |Q| = O ( Z n n ( − q +( m − s ) ) , (6.7) whenever j ∈ [ m ], P ∈ O , and Q is a collection of dyadic cubes Q ∈ Q mdn such thateach Q ∈ Q satisfies Q ⊂ A n , Q ∩ Z ( P ) = ∅ , and π j ( Q ′ ) = π j ( Q ) for all Q, Q ′ ∈ Q .Here (slightly abusing notation) we denote A n = supp( µ n ) = A n, ×· · ·× A n,m , where A n,i are the steps in the construction of each of the independent fractal percolations.To that end, we fix such j ∈ [ m ], P ∈ e O and collection Q , and adapt theestimation (5.15) to the present setting as follows: |Q| − nmd = L md ( ∪ Q ∈Q Q ) = O (1) Z y ∈ B H md − q ( P − ( y ) ∩ ∪ Q ∈Q Q ) d L q ( y )= O (1) Z y ∈ B Z z ∈ B H ( m − d − q ( P − ( y ) ∩ A n ∩ { x j = z } ) d H d ( z ) d L q ( y ) , (6.8)where B = B (0 , O (2 − n )) ⊂ R q and B = π j ( Q i ) ⊂ R d . Here, the first estimatefollows from Proposition 6.2 applied to the trivial polynomial and f := P (recallthat P ∈ P reg for P ∈ O ). The integration is over B (0 , O (2 − n )) because the P ∈ O are uniformly Lipschitz and Q ∩ P − (0) = ∅ for all Q ∈ Q . The secondestimate in (6.8) follows from Proposition 6.2 applied to π j , using the hypothesis oftransversality with respect to the planes { x j = 0 } .From now on we assume that n is large enough that B ⊂ B (0 , δ ), where δ is thenumber in the definition of e O . Note that H ( m − d − q (cid:0) P − ( y ) ∩ A n ∩ { x j = z } (cid:1) ≤ n ( m − s − d ) e Y e Pn , for e P ∈ e O defined by e P ( x , . . . , x m − ) = P ( x , . . . , x j − , z, x j , . . . , x m − ) − y . Weinfer from (6.8) that |Q| − nmd ≤ O (1)2 − nq − nd n ( m − s − d ) Z n , from which (6.7) follows.After this, the rest of the proof proceeds just like the proof of Theorem 5.2, usingthat ν (1) n × · · · × ν ( m − n is a WSDM with respect to O . (cid:3) Remarks . (i) In general, one cannot remove the requirement that the tangentsof P − (0) are transversal to the planes { x i = 0 , i ∈ I } , even if they fail atonly finitely many points. To see this, let us consider the following example:Let A = A perc(1 ,p ) ⊂ [0 ,
1] with < s < , µ n = ν perc(1 ,p ) n × ν perc(1 ,p ) n andconsider the random variables Y Pn for the quadratic polynomials P ( x, y ) = | ( x, y ) − ( x , y ) | − λ , for x , y ∈ [0 , λ >
0. Thus, we are intersecting A × A and ν × ν with circles. Then, for ν × ν almost every ( x, y ) ∈ A × A , andall circles P − (0) through ( x, y ) with a vertical tangent at ( x, y ), it holds that Y Pn −→ ∞ as n → ∞ . Let us give a short sketch of the proof: The circulararc P − (0) ∩ π − ( I ) has length Ω(2 − n/ ), where I ∈ Q n is the dyadic interval ATTERNS IN RANDOM FRACTALS 49 containing x . If I ′ ∈ Q k is the dyadic interval containing y with k ∼ n/
2, then P − (0) typically intersects roughly 2 ns/ squares I × I ′′ ∈ Q n , where I ′′ ⊂ I ′ .Thus, Y Pn is bounded from below by Ω(2 n (1 − s ) × ns/ × − n ) = Ω(2 n (1 − s/ )and this grows at an exponential rate as n → ∞ , whenever s < / P − (0) cannot be weakened to the maps P
7→ H md − q ( P − (0) ∩ Q ) being H¨older continuous for Q ∈ Q md .(ii) On the other hand, there is scope to weaken the transversality assumptionwith respect to coordinate hyperplanes, and it might even be possible to elimi-nate this assumption altogether by considering a suitable version of the dyadicmetric (adapting the proof of Corollary 5.9). To avoid excessive technicalities,and since this is not an issue in our main applications, we do not pursue this.(iii) The method behind the proof of Theorem 6.3 is not tied to algebraic surfacesand it also works for other parametrized families of smooth surfaces satisfyinganalogous transversality and dimensional conditions. Since all our applicationsregarding the existence of various configurations in A will be derived by inter-secting the Cartesian powers of A with the elements of some P reg r,q,M , we shallnot discuss these generalizations here.In this setting, the analogue of Lemma 5.7 also holds: Lemma 6.7.
In the setting of Theorem 6.3, if s > q/m , then P ( Y P > > .Proof. The proof is nearly identical to the proof of Lemma 5.7. The geometry ofthe planes V only enters the proof via the estimate H k ( V ∩ Q ) = O (2 − nk ) for Q ∈ Q mdn , and the cardinality bound (5.20). In the present context, we clearly have H md − q ( Z P ∩ Q ) = O (2 − n ( md − q ) ) for Q ∈ Q mdn . With respect to (5.20), we note thatin Lemma 5.7 this is obtained by showing that V ( √ d − n ) ∩ e π − ( e Q ) ∩ Q j can becovered by a suitable parallelepiped. If we replace V by Z P then the same holdsexcept that instead of the linear parallelepiped, we need to consider an O (2 − n )-neighbourhood of the variety Z ( P ) ∩ e π − ( x ) for some fixed x ∈ e π ( Q ) and use thecoarea formula (Proposition 6.2) to verify the volume argument. Details are left tothe interested reader. (cid:3) Non-linear configurations.
In order to find non-linear configurations insidethe fractal percolation set, we will apply the following localized version of Theorem6.3.
Corollary 6.8.
Let P ∈ P r,q,md and let a = ( a , . . . , a m ) ∈ R md be a regular pointfor P such that P ( a ) = 0 and a i = a j for all i = j . Let V be the tangent plane to P − (0) at a . Suppose that for each index set I ( [ m ] , and each i ∈ [ m ] \ I , k ∈ [ d ] , dim V ∩ H I = max { , ( m − | I | ) d − q } , dim V ∩ H I,i,k = max { , ( m − | I | ) d − q − } . Further, let s > q/m .Then there is a neighbourhood O of P such that P ( Z P ∩ A m = ∅ for all P ∈ O ) > , where A = A perc ( d,p ) is the fractal percolation set of dimension s .Proof. Since V is transversal to coordinate hyperplanes, by replacing a by a nearbypoint we may assume that a is not on the boundary of any dyadic cube. Pick n large enough that • The dyadic cubes Q i ∈ Q dn containing a i are all disjoint (this is possiblesince a / ∈ ∆), • The transversality conditions continue to hold if V is replaced by the tangentplane at y to { P = P ( y ) } for all y ∈ Q × · · · × Q m (this is possible bycontinuity).Let f i , i ∈ [ m ] be the homotheties mapping [0 , d to Q i , and let ν ( i ) n = f − i ν n | Q i bethe restriction of the original process to the cubes Q i , rescaled back to the unit cube.Let us condition on the cubes Q , . . . , Q m surviving, so that ν ( i ) n become multiplesof independent copies of ν perc( d,p ) . Now consider the polynomial P ( x , . . . , x m ) = P ( f ( x ) , . . . , f m ( x m )) . We have set things up so that P satisfies the conditions of Theorem 6.3, whichwe can then apply to ν ( i ) n as above. An application of Lemma 6.7 concludes theproof. (cid:3) The next straightforward lemma provides a convenient way to verify the transver-sality assumptions in the last corollary in the case when q ≤ d . Lemma 6.9.
Let P ∈ P r,q,md and let a = ( a , . . . , a m ) ∈ R md . If either of thefollowing conditions hold, then P and a satisfy the assumptions of Corollary 6.8.(1) d = q , and for each j ∈ [ m ] , the vectors ( ∂P i /∂x j ( a )) qi =1 are linearly inde-pendent.(2) d > q , and for each j ∈ [ m ] and k ∈ [ d ] , the vectors ( ∂P i /∂x j ( a )) qi =1 togetherwith the canonical vector e k = (0 , . . . , , . . . , ∈ R d are linearly independent.Proof. Fix I ( [ m ], let L = DP ( a ) and L ( x ) = ( x i ) i ∈ I . The tangent plane to { P = P ( a ) } at a is ker( L ), and ker( L ) = { x i = 0 , i ∈ I } . Note also that forthe map L ( x ) = ( L x, L x ), ker( L ) = ker( L ) ∩ ker( L ). Hence, transversality ATTERNS IN RANDOM FRACTALS 51 with respect to ker( L ) follows if L has full rank q + d | I | . Suppose then that thereis a non-trivial linear combination among the rows of L and L . Since the rowsof L are clearly linearly independent, some row of L has a non-zero coefficient.Pick j ∈ [ m ] \ I . Since projections respect linear dependency, there is a non-trivial linear combination among the rows L , L corresponding to x j , but for L these rows consist of 0’s, so ( ∂P i /∂x j ( a )) qi =1 is linearly dependent, contradicting theassumption. Observe that taking I = ∅ we get that a is a regular point of P .Note that transversality with respect to the planes { x kℓ } for ℓ ∈ I follows fromthe above considerations. It remains to check transversality also with respect to theplanes of the form { x kℓ = 0 } where ℓ / ∈ I . If | I | < m −
1, then the same argumentas above applies, taking j / ∈ I ∪ { ℓ } . If I = m − d = q , then we already haveker( L ) = 0 and we are done. Finally, if | I | = m −
1, and q > d , we apply the sameargument as above with j / ∈ I , using that linear independence still holds when acanonical vector is added. (cid:3) We now provide several applications of Corollary 6.8 to the existence of variousgeometric configurations inside A perc . We start with a small extension of a Theoremof Rams and Simon [31] asserting that the set of distances between points of A perc has non-empty interior whenever s > ; we show that in this case A perc containsall sufficiently small distances. This stronger form can be easily deduced from theresult of Rams and Simon together with Lemma 3.1; we give an alternative proofusing Theorem 6.3 because the proof is a model case for the later applications, andis also a simple example of a situation where the linear result, Theorem 5.2, cannotbe directly used. Corollary 6.10.
Suppose s > , and write D ( A ) = {| x − y | : x, y ∈ A } , where A = A perc ( d,p ) . Then:(1) Almost surely, there is ε > such that (0 , ε ) ⊂ D ( A ) .(2) For any ε > , P (cid:16) (0 , √ d − ε ) ⊂ D ( A ) (cid:17) > .Proof. For each λ >
0, let P λ ( x, y ) = | x − y | − λ ∈ P reg2 , , d .If λ ∈ (0 , √ d ), we can find x , y ∈ [0 , d such that | x − y | = λ and the d coordinates of x − y are all non-zero. A calculation shows that the hypotheses ofCorollary 6.8 for P λ and ( x , y ) are met, so there exists an interval I λ around λ such that P ( I λ ⊂ D ( A )) > . By Harris’ inequality (Lemma 3.3), this implies that e.g. P ([1 / , / ⊂ D ( A )) =: c ( d, p ) > . The zero-one law from Lemma 3.1 then yields the first claim, while the second claimfollows using the first and Harris’ inequality once again. (cid:3)
Next, we show that A perc( d,p ) contains simplices of all small positive volumes,provided s > / ( d + 1). Related to this, in [12, Theorem 3.7] (see also [15]) it isshown that for some explicit ε d >
0, if A ⊂ R d has Hausdorff dimension d − ε d ,then the simplices determined by d + 1 points in A determine a positive measureof volumes. Again, by considering the special case of fractal percolation, we get asharp bound for this phenomenon. Corollary 6.11.
Suppose s > d +1 . Then a.s. there exists ε > such that A = A perc ( d,p ) contains d + 1 points determining a simplex of volume v for each v ∈ (0 , ε ) .Proof. Given v >
0, let P v ∈ P d, ,d ( d +1) be given by P v ( x , . . . , x d +1 ) = det( M ( x , . . . , x d +1 )) − d ! v , where M ( x , . . . , x d +1 ) is the matrix with columns ( x , , . . . , ( x d +1 , P v ( x ) =0, then the simplex with vertices x i has volume v (the reciprocal is not true sincethe determinant gives the oriented volume, so it could be − v as well). Notethat ∂P/∂x ki ( z , . . . , z d +1 ) is ± the ( d − e π k ( z j ) , j ∈ [ m ] \ { i } , where e π k ( x , . . . , x d +1 ) = ( x , . . . , x k − , x k +1 , . . . , x d +1 ). Henceif z , . . . , z d +1 are generic points in ]0 , d , then Lemma 6.9, and hence Corollary6.8, apply to P v and t v z = ( t v z , . . . , t v z d +1 ) where t v ∈ (0 ,
1) is chosen so that P v ( t v z ) = 0 (after possibly permuting the z i to ensure det( M ( z )) > (cid:3) Now, we look at congruent copies of triangles in R . Corollary 6.12.
Suppose s > . Then for every triple z = ( z , z , z ) of non-collinear points in ]0 , , there exists a neighbourhood O of z such that with positiveprobability, A perc (2 ,p ) contains an isometric copy of { y , y , y } whenever ( y , y , y ) ∈O .Proof. Given y = ( y , y , y ) ∈ ( R ) , let P y ( x , x , x ) = ( | x − x | − | y − y | , | x − x | − | y − y | , | x − x | − | y − y | ) . Then P y ∈ P , , , and if P y ( x , x , x ) = 0, then { x , x , x } is isometric to { y , y , y } .Note that Lemma 6.9 is not applicable since q > d in this case, but one can di-rectly verify the assumptions from Corollary 6.8 as follows. Firstly, DP z ( z ) hasfull rank. By rotating the given z , z , z , we may assume z i − z j is not in a coor-dinate line for i = j . Then for any j ∈ [3] and any k ∈ [2], one can verify thatdim(ker( DP z ( z )) ∩ { x ki = 0 } ) = 2, and ker( DP z ( z )) ∩ { x i = 0 } is a one-dimensionalsubspace not contained in a coordinate hyperplane. We can then apply Corollary6.8 to obtain the desired statement. (cid:3) ATTERNS IN RANDOM FRACTALS 53
Remark . The proof of Corollary 6.12 does not extend to simplices (or otherpolyhedra) in higher dimensions, since one can see that the transversality conditionsfail. Geometrically, the reason is the following: to find an isometric copy of a simplexof vertices z , . . . , z d +1 (with the same vertex order), the first coordinate y is free,but y is constrained to lie in the sphere of center y and radius | z − z | , so once y is fixed, the angle of the corresponding tangent plane with the coordinate plane y = 0 is zero. More precisely, the transversality of ker DP (for the correspondingpolynomial P ) fails with respect to the planes H I , when | I | = 2.This issue does not arise in R because once the second vertex is fixed, the thirdvertex (and therefore the full triangle) is completely determined. Formally, thismeans that dim ker( DP z ( z )) ∩ H I = 0 whenever | I | ≥ Scale-invariant patterns.
So far, in § all the corresponding configurations (distances,volumes or triangles) in the fractal percolation limit set. We now turn to a class ofconfigurations which are scale, and indeed homothety-invariant. Thanks to Corol-lary 3.2, for this kind of patterns we will be able to obtain more pleasant results:a.s. A perc( d,p ) will contain all the configurations in each class.We start by considering the angles determined by A perc( d,p ) . The problem of whatlower bound on the Hausdorff dimension ensures that a subset of R d contains a givenangle was investigated in [27, 16] and is far from settled. For fractal percolation, weget the following result: Corollary 6.14.
Suppose s > . Then, almost surely, all angles in ]0 , π [ can beformed by three points in A perc ( d,p ) .Proof. For each λ ∈ ] − , P λ ( x , x , x ) = (( x − x ) · ( x − x )) − λ | x − x | | x − x | ∈ P , , d . Note that x = ( x , x , x ) ∈ R d satisfies cos( ∠ x x x ) = λ if and only if P λ ( x ) = 0.Fix λ ∈ ] − ,
1[ and pick a point x ′ = ( x ′ , x ′ , x ′ ) ∈ ]0 , d such that P λ ( x ′ ) = 0and, moreover, x ′ i − x ′ j does not lie in a coordinate plane for i = j . Then Lemma6.9 can be used to show that P λ and x ′ satisfy the hypotheses of Corollary 6.8.We deduce that there is an open interval I λ containing cos − ( λ ) such that, withpositive probability, A perc( d,p ) contains all angles in I λ . By the zero-one law forfractal percolation (Corollary 3.2), ‘positive probability’ can be replaced by ‘fullprobability’. Finally, covering ]0 , π [ by countably many such intervals I λ we reachthe final conclusion. (cid:3) Remark . The extreme angles α ∈ { , π } correspond to three collinear points in A . For the existence of three points on a line in A perc , the threshold is s > ( d − / which can be seen by applying Lemma 5.7 to the plane V = { ( x, y, z ) ∈ ( R d ) : z = x + λ ( y − x ) for some λ ∈ R } ∈ A d, d +1 .Another problem that has received a lot of attention is: what geometric conditionson a subset A ⊂ R d ensure that A contains the vertices of an equilateral triangle(or, more generally, a similar copy of a given triangle)? In particular, does any lowerbound on the Hausdorff dimension of a set A ⊂ R d suffice? For d = 2, the answeris no: there exists sets A ⊂ R of Hausdorff dimension two which do not containthe vertices of an equilateral triangle, see [9], [26]. However, the answer is yes, forsimilar copies of any triangle, if one additionally assumes a Fourier decay conditionon a Frostman measure on A [2], although even in this case the lower bound onthe dimension is not explicit. Recently, A. Iosevich and B. Liu [18] showed thatfor d ≥ ε d > A ⊂ R d of Hausdorffdimension > d − ε d does contain the vertices of an equilateral triangle. The problemis open in dimension d = 3. For fractal percolation sets, we have the followingresult: Corollary 6.16.
Suppose s > / . Then almost surely, A = A perc ( d,p ) contains thevertices of a triangle similar to an arbitrary non-degenerate triangle in R d .Proof. Given a, b > a + b >
1, let P a,b : R d → R be given by P a,b ( x , x , x ) = ( | x − x | − a | x − x | , | x − x | − b | x − x | ) . Then P a,b ( x , x , x ) = 0 if the triangle ( x , x , x ) is similar to a triangle withside-lengths 1 , a, b . Fix, then, a, b as above, and let y , y , y ∈ R d be such that P a,b ( y , y , y ) = 0 and y i − y j does not lie in a coordinate hyperplane for i = j . Wecan then apply Lemma 6.9, Corollary 6.8 and conclude the proof as in Corollary6.14. (cid:3) For similar reasons to those explained in Remark 6.13, our methods do not directlyapply to the problem of finding similar copies of higher-dimensional polyhedra inside A perc( d,p ) . However, in the plane, we can extend Corollary 6.16 to general polygons: Corollary 6.17. If s > − /m , then a.s. A perc (2 ,p ) contains a similar copy of all m -point sets { a , . . . , a m } such that no three of the a i are collinear.Proof. The starting point of the proof is the following observation: Suppose that { x ′ , . . . , x ′ m } is a similar copy of { a ′ , . . . , a ′ m } such that x ′ j = h ( a ′ j ) for all j ∈ [ m ]and some h ∈ SIM . Then there are small neighbourhoods U of x ′ = ( x ′ , . . . , x ′ m )and e U of a ′ = ( a ′ , . . . , a ′ m ) such that for all ( a , . . . , a m ) ∈ e U , x ∈ U yields a similarcopy of a ′ if | x j − x i | / | x − x | = | a j − a i | / | a − a | whenever 3 ≤ i ≤ m and j ∈ { , } . ATTERNS IN RANDOM FRACTALS 55
Now to the details. Let N = { ( i, j ) : 3 ≤ i ≤ m , j ∈ { , }} . For any a = ( a , a , . . . , a m ) ⊂ ( R ) m , where no three of the { a , . . . , a m } arecollinear, denote a ( i, j ) = | a j − a i | / | a − a | , for all ( i, j ) ∈ N , and define P i,j,a ( i,j ) ( x , . . . , x m ) = | x j − x i | − a ( i, j ) | x − x | for ( x , . . . , x m ) ∈ ( R ) m . Finally, consider the polynomial P a = ( P i,j,a ( i,j ) ) ( i,j ) ∈ P , m − , m . Fix a ′ ∈ (]0 , d ) m \ ∆. As we have observed, if x ′ is close to a ′ and P a ′ ( x ′ ) = 0,then x ′ is similar to a ′ . Our goal is to verify that the assumptions of Corollary 6.8apply to P a ′ and a ′ . We cannot apply Lemma 6.9 directly, but we argue analogouslyas follows: let 1 ≤ i < j ≤ m . It is enough to show that the rows of DP a ′ ( a ′ )together with the vectors e ki , e kj , k ∈ { , } , are linearly independent (and hencea basis). Here e ki is the canonical vector for the coordinate x ki . Suppose, on thecontrary, that there is a non-trivial linear combination. Since e ki , e kj are linearlyindependent, some ∇ P i,j,a ′ ( i,j ) ( a ′ , . . . , a ′ m ) must have a non-zero coefficient. Pick ℓ ∈ { , , i, j } \ { i , j } . Then there is a non-trivial linear combination among ∂P i,j,a ′ ( i,j ) /∂x ℓ ( a ′ , . . . , a ′ m ), but a calculation (using the non-collinearity of the a ′ i )shows this is not the case.Hence Corollary 6.8 can be applied, and together with Corollary 3.2 and Lemma3.3 this concludes the proof. (cid:3) Remark . A straightforward adaptation of the proof of Proposition 5.12, onecan again see that the dimension thresholds provided by Corollaries 6.10–6.17 aresharp for packing dimension even for deterministic sets (up to the endpoint).7.
Patterns in sets of positive measure
The dimension of intersections and patterns.
We refine the results inSections 5–6 by providing upper and lower bounds for the dimension of intersectionsof A m with algebraic varieties. As a direct corollary, this yields dimension estimatesfor the ‘number of times’ a given configuration is found in the set A . Theorem 7.1.
Let A = A perc ( d,p ) . In the setting of Theorem 6.3, if s > q/m , thereis a neighbourhood O of P such that:(1) For every δ > there is a finite random variable K such that A m ∩ Z P \ ∆( δ ) can be covered by K n ( ms − q ) cubes in Q mdn for all P ∈ O . In particular, a.s. dim P ( A m ∩ Z P \ ∆) ≤ ms − q for all P ∈ O . (2) For any δ > , with positive probability, dim H ( A m ∩ Z P \ ∆) > ms − q − δ for all P ∈ O .Proof. To begin, we can partition [0 , md \ ∆ into a countable union of products Q := Q × . . . × Q m , where Q i ∈ Q dn for some (variable) d and ( π j ( Q i )) mi =1 are disjointfor all j . By considering the restriction to each of these cubes, and applying theusual independence and rescaling arguments, we may replace A m by the product of m independent fractal percolations (for the upper bound we are using the countablestability of packing dimension). Let A ′ denote the product of the m copies of A perc( d,s ) .Write µ n = ν (1) n × · · · × ν ( m ) n with ν ( i ) n independent realizations of ν perc( d,p ) n . Toestablish the first claim, suppose that Z P intersects M disjoint cubes Q ∈ Q mdn forsome P ∈ O . Write P y = P − y , and note this is in O for small enough y . By thecoarea formula (Proposition 6.2), using that the d -Jacobian of P y is bounded awayfrom zero for y small, we deduce that if n is large enough, then M − nms ≤ O (1) Z y ∈ B (0 ,O (2 − n )) ⊂ R q Y P y n d L q ≤ O ( K − nq ) . (7.1)This shows that M ≤ O K (1)2 n ( ms − q ) , so that the packing (and indeed the box-counting) dimension of Z P is at most ms − q for all P ∈ O , establishing the firstclaim.For the second claim, fix t ∈ (0 , md ) and let e A = e A perc( md,q ) be fractal percolationon R md of Hausdorff dimension md − t , independent of µ n , with correspondingapproximating measures e µ n . Now set µ n = µ n e µ n . The sequence µ n is neither fractal percolation nor a product of independent percola-tions. However, it is easily checked to be a martingale measure, and the dependencystructure matches exactly that of the product of independent copies µ n : there aredependencies only among coordinate directions. In particular, µ n converges almostsurely to a measure µ supported on A ′ ∩ e A , which has dimension ms − t . The proof ofTheorem 6.3 apply verbatim to µ , while the proof of Lemma 6.7 (or rather Lemma5.7) extends with very minor changes (the estimates for the L norm actually getbetter since there is more independence than in the setting of Lemma 6.7). Wededuce that if ms − t > q , then P (cid:16) ( A ′ ∩ Z P ) ∩ e A = ∅ for all P ∈ O (cid:17) > , where O is a neighbourhood of P which is independent of s and t (so long as ms − t > q ). Since A ′ and e A are independent, this implies that there are positivelymany realizations of A ′ such that the above holds with positive probability with ATTERNS IN RANDOM FRACTALS 57 respect to the construction of e A . Fixing such a realization of A ′ , Theorem 3.5allows us to conclude that dim H ( A ′ ∩ Z P ) ≥ t for all P ∈ O . Since t < ms − q isarbitrary, we get the second claim. (cid:3) Remarks . (i) The same result holds in the setting of Theorem 5.2 (with k inplace of md − q ), either by noting that the proof works works verbatim in thatcase, or by seeing Theorem 5.2 as (essentially) the particular case of Theorem6.3 in which the polynomial P is affine.(ii) It seems very likely that using the approach of [36, Theorem 12.8], one canimprove the second part of the theorem as follows: almost surely, for every P ∈ O such that Y P > H ( A ′ ∩ Z P \ ∆) ≥ ms − q. Recall that Y P > P ’s. For the sake of simplicity, we willwork with the slightly weaker version above.For a fixed P , we also have the following upper bound, without any transversalityassumptions: Lemma 7.3.
Let P ∈ P reg r,md,q , and let A = A perc ( d,p ) . Assume s > q/m . Thenalmost surely, for any δ > , dim B ( Z P ∩ A m \ ∆( δ )) ≤ ms − q. Proof.
We use the first moment method. Let K n be the number of cubes Q ∈ Q mdn such that Q ∩ A mn ∩ Z P \ ∆( δ ) = ∅ . For large n (in terms of δ ) each cube Q ⊂ [0 , md \ ∆( δ ) survives in A mn with probability 2 nm ( s − d ) , so for each ε > P ( K n > n ( ms − q + ε ) ) ≤ − nε n ( q − ms ) E ( K n ) = O (2 − εn ) , note that Z P intersects O (2 n ( md − q ) ) cubes in Q mdn . By the Borel-Cantelli Lemma,a.s. K n ≤ n ( ms − q + ε ) for all large n , which gives the claim. (cid:3) As a corollary of Theorem 7.1, we can now prove Theorem 1.3:
Theorem 7.4. If s > d − ( d + 1) /m , then a.s. for each m point set S ⊂ R d , dim (cid:8) ( a, b ) ∈ R × R d : aS + b ⊂ A (cid:9) = m ( s − d ) + d + 1 , where dim is either Hausdorff or packing dimension.Proof. By covering the parameter space by countably many neighbourhoods O ,we can work with a fixed O for which the conclusions of Theorem 7.1 hold. Theupper bound is a direct consequence of the first part of Theorem 7.1. For the lowerbound, we apply the second part of Theorem 7.1 to each δ >
0, use Corollary 3.2 toupgrade positive probability to full probability, and then let δ ↓
0. As in the proof of Corollary 5.9, we need to consider the dyadic metric to bypass the failure oftransversality with respect to the coordinate hyperplanes for certain patterns. (cid:3)
Remark . Similar results hold for other classes of configurations. For patternswhich are not scale-invariant, we cannot in principle use the zero-one law to getrid of the δ in the lower bound, but see Remarks 7.2(ii) for a possible approachto overcome this. If transversality with respect to coordinate hyperplanes fails atsome points, then we do not in get a uniform upper bound for the dimension (butwe do for each given configuration, thanks to Lemma 7.3). However, as pointedout in Remarks 6.6(ii), it may be possible to remove the assumption of hyperplanetransversality altogether.7.2. Lack of patterns in sets of full measure.
As an application of Lemma 7.3,we show that whenever s < qm − it is possible to find a full ν -measure subset A ′ ⊂ A such that ( A ′ ) n ∩ Z P = ∅ . We will show in the next section that the oppositehappens when s > qm − . Proposition 7.6.
Let ν = ν perc ( d,p ) and A = A perc ( d,p ) . Let ν = ν perc ( d,p ) and A = A perc ( d,p ) . If P ∈ P reg r,q,md and s = s ( d, p ) < qm − , then a.s there is a Borel set A ′ ⊂ A of full ν -measure such that A ′ does not contain distinct points x , . . . , x m with P ( x , . . . , x m ) = 0 .Proof. We know from Lemma 7.3 that a.s.dim P ( Z P ∩ A m \ ∆) ≤ ms − q < s using the assumption s < qm − for the second inequality. Let A ′ = A \ π ( Z P ∩ A m \ ∆) . Since A \ A ′ has dimension < s , we have ν ( A ′ ) = ν ( A ). On the other hand, it isclear from the definition that A ′ cannot contain distinct points x , . . . , x m such that P ( x , . . . , x m ) = 0. (cid:3) Assuming the same transversality conditions as in Corollary 6.8, the previousProposition extends to the more delicate case of the threshold s = q/ ( m − Theorem 7.7.
Let ν = ν perc ( d,p ) and A = A perc ( d,p ) . If P ∈ P reg r,q,md satisfies theAssumptions of Corollary 6.8 and s = s ( d, p ) = qm − , then a.s. there is a Borel set A ′ ⊂ A of full ν -measure such that A ′ does not contain distinct points x , . . . , x m with P ( x , . . . , x m ) = 0 .Proof. As in the proof of Proposition 7.6, it is enough to show that a.s. ν ( π ( Z P ∩ A m \ ∆)) = 0 . (7.2) ATTERNS IN RANDOM FRACTALS 59
Covering R md \ ∆ by cubes Q ∈ Q mdn , n ∈ N , with Q ∩ ∆ = ∅ and conditioning on Q ⊂ A mn , we see that (7.2) is implied by ν ( π ( A (1) × · · · × A ( m ) ∩ Z P )) = 0 , (7.3)where A (1) , . . . , A ( m ) are independent fractal percolations, and ν = ν (1) is the fractalpercolation measure on A (1) .Given a ∈ R d , define e P a ∈ P r,q, ( m − d by e P a ( x , . . . , x m ) = P ( a, x , . . . , x m ) . We claim that for each large enough n and each Q ∈ Q dn , there is a set { a , . . . , a ℓ } with ℓ = O P (1), such that A (2) n × · · · × A ( m ) n ∩ π ( π − ( Q ∩ Z P )) = ∅ only if A (2) n × · · · × A ( m ) n ∩ Z ( e P a i ) = ∅ for some a i ∈ { a , . . . , a ℓ } . To that end, let Q ′ ⊂ A (2) n × · · · × A ( m ) n , Q ′ ∈ Q ( m − dn and denote by x ′ the center point of Q ′ . Suppose Q ′ ∩ π ( π − ( Q ∩ Z P )) = ∅ so that P ( x , x , . . . , x m ) = 0 for some x ∈ Q , ( x , . . . , x m ) ∈ Q ′ . Note that P ( x , x ′ ) = O P (2 − n ). Letting y = G ( P , P , x ), where P i : R d → R q , P ( x ) = P ( x, x ′ ) − P ( x , x ′ ), P ( x ) = P ( x, x ′ ), and G is as in Lemma 6.1, we have that P ( y, x ′ ) = 0 and | y − x | = O P (2 − n ). Here the use of Lemma 6.1 is justified becausethe tangent planes of Z P are uniformly transversal with respect to the coordinateplane π ( R md ).Let C be a sufficiently large constant to be chosen later, and let { a , . . . , a ℓ } be a maximal C − − n separated subset of CQ (the cube with the same centreas Q and side length C − n ). Pick a ∈ { a , . . . , a ℓ } such that | y − a | ≤ C − − n and let z = e G ( e P , e P , x ′ ), where e P , e P : R ( m − d → R q , e P ( x ) = P ( y, x ), e P ( x ) = P ( a, x ) = e P a ( x ) and e G is again provided by Lemma 6.1 (now using the transversalitywith respect to the plane π ( R md )). We conclude that z ∈ Z ( e P a ) and | z − x ′ | = O P ( C − − n ) < − n − so that z ∈ Q ′ provided C is large enough depending on P .Thus, if C = O P (1) is large enough, then { a , . . . , a ℓ } is the desired family.Now fix n ≫ Q ∈ Q dn , and let F Q denote the event A (2) n × · · · × A ( m ) n ∩ π ( π − ( Q ∩ Z P )) = ∅ . Then F Q is independent of the realization of ( A (1) n ) n , and P ( F Q ) ≤ ℓ X i =1 P ( A (2) n × · · · × A ( m ) n ∩ Z ( e P a i ) = ∅ ) . Note that Z ( e P a ) can be covered by O (1)2 n (( m − d − q ) cubes in Q n for a ∈ [0 , d (withthe O constant independent of a ); the proof is the same as (7.1), together with compactness. Since s = q/ ( m − a ∈ [0 , d , P ( A (2) n × · · · × A ( m ) n ∩ Z ( e P a ) = ∅ ) ≤ q n for some sequence q n ↓ a . We deduce that P ( F Q ) = O ( q n ). Let uswrite e A = A (1) × · · · × A ( m ) . Observe that E ( ν ( Q ∩ π ( e A ∩ Z P ))) ≤ P ( F Q ) E ( ν ( Q )) = O ( q n ) E ( ν ( Q )) , using the independence of ν and F Q . Since E ( k ν k ) = 1, we conclude that E ( ν ( π ( e A ∩ Z P ))) = X Q ∈Q dn E ( ν ( Q ∩ π ( e A ∩ Z P ))) = O ( q n ) . Since n ≫ (cid:3) Patterns in sets of positive measure.
Having obtained the critical dimen-sion for the existence of different patterns in fractal percolation, we next turn ourattention to the following problem: what is the critical value s c , such that if s > s c then all ‘large’ subsets A ′ ⊂ A (i.e. the ones with ν ( A ′ ) >
0) contain the requiredpattern? As described in the introduction, this question is motivated by variousanalogous results in the discrete setting, in particular the Green-Tao and Conlon-Gowers-Schacht theorems on the existence of arithmetic progressions inside positivedensity subsets of the primes and random discrete sets.It turns out that the answer to this question is closely related to the dimensionof the pattern structures investigated earlier in this section. Let us consider theproblem of finding homothetic copies of a set S for concreteness. As explained inthe proof of Proposition 7.6, if the dimension of { ( λ, x ) : x + λS ⊂ A } is < s , thenthe points in A that belong to at least one such x + λS also lie in a set of dimension < s , so we can remove all such points to end up with a full ν -measure subset of A which contains no homothetic copy of S . On the other hand, if dim H { ( λ, x ) : x + λS ⊂ A } > s , then we will show that, almost surely, all subsets A ′ ⊂ A withpositive ν measure do contain a homothetic copy of S . Indeed, we have the followingabstract result for polynomial configurations: Theorem 7.8.
Let Λ be an open subset of R M for some M ∈ N . Suppose λ P λ is a continuous map from Λ to P r,q,md such that for each λ ∈ Λ , the polynomial P λ satisfies the hypotheses of Corollary 6.8 (for some point a = a ( λ ) ).Suppose s ( d, p ) > qm − . Then a.s. there is a nonempty open set U ⊂ Λ such thatfor all Borel sets A ′ ⊂ [0 , d with ν ( A ′ ) > there exist r > and t ∈ R d such thatfor all λ ∈ U there are points x , . . . , x m ∈ A ′ with P λ ( rx + t, . . . , rx m + t ) = 0 .If, furthermore, P λ ( x , . . . , x m ) = 0 if and only if P λ ( rx + t, . . . , rx m + t ) = 0 for all λ ∈ Λ , r > and t ∈ R d , then a.s. for all λ ∈ Λ and all Borel sets A ′ with ν ( A ) ′ > there are distinct points x , . . . , x m ∈ A ′ such that P λ ( x , . . . , x m ) = 0 . ATTERNS IN RANDOM FRACTALS 61
Combining this with (the proofs of) Corollaries 5.9, 6.10, 6.11, 6.12, 6.14, 6.16,6.17, we immediately obtain Theorem 1.4.We note that the threshold for s is sharp in a rather strong way, thanks toTheorem 7.7. For non-scale invariant patterns, Theorem 7.7 has to be applied toa countable family of polynomials that witnesses a dense set of patterns in theappropriate family. For example, if s ∈ (1 / , A contains all smalldistances a.s., there exists a set A ′ ⊂ A of full ν -measure which does not containany rational distances. Remark . Note that in Corollary 5.11, we have k = d so that s < md − km − for all s ≤ d . So there is no relative Szemer´edi theorem for translated copies.We give the idea of the proof of Theorem 7.8 in the special case of homotheticcopies. Suppose that s > d − m − , and for T = { t , . . . , t m − , } , let V T be as inthe proof of Corollary 5.9, and let Γ be the family of all such planes. Then, anapplication of Theorem 5.2 to ( ν n ) m − and R (Γ) implies that for any fixed Q ∈ Q dn ,there cannot be more than O (2 n (1+( m − s − d )) )cubes Q ′ ∈ Q mdn , Q ′ ⊂ ( A n ) m , Q ′ ∩ V T = ∅ with π j ( Q ′ ) = Q . Note that the O -constant is random but uniform in n and T . Recalling that ( A n ) m ∩ V T intersectsroughly 2 n ( d +1+ m ( s − d )) cubes in Q mdn , we observe that in order to violate ( A n ) m ∩ V T = ∅ , we need to remove at least Ω(2 ns ) cubes from A n . Letting n → ∞ , this essentiallyshows that ( A ′ ) m ∩ V T = ∅ whenever A ′ ⊂ A has full µ -measure. Finally, to passfrom full ν -measure to ν ( A ′ ) >
0, we will use a density point argument.We pass to the details. For F ⊂ [0 , d , let N n ( F ) denote the the number of cubes Q in Q dn such that Q ∩ F = ∅ . Recall also the notation N n := N n ( A ) from Section3. For the rest of this section, we assume that Λ, λ → P λ satisfy the assumptionsof Theorem 7.8, and s > qm − . Moreover, we always denote ν = ν perc( d,p ) and A = A perc( d,p ) with s = s ( d, p ). Proposition 7.10.
Fix λ ∈ Λ . There are ε > , n ∈ N , a cube Q ∈ Q mdn , andan open neighbourhood U ⊂ Λ of λ (all deterministic) such that the following eventhas positive probability: for any compact A ′ ⊂ A such that ( A ′ ) m ∩ Z ( P λ ) ∩ Q = ∅ for some λ ∈ U , we have ν ( A \ A ′ ) ≥ ε .Proof. In the course of the proof, C i , C ′ i denote finite positive constants independentof n or any dyadic cubes. By Lemma 6.7 and (the proof of) Corollary 6.8, there arean open neighbourhood U of λ and a small dyadic cube Q such that, with positiveprobability, Z Q ∩Z ( P λ ) ( ν n ) m d H md − q ≥ C for all λ ∈ U, n ∈ N . (7.4) Indeed, one only needs to take Q to be a small enough dyadic cube containing a ( λ ) and disjoint from ∆, such that the transversality conditions hold on Q for all λ ∈ U , as in the proof of Corollary 6.8.In particular, if N n,λ = N n,λ ( A n ) is the number of cubes Q ∈ Q mdn such that Q ⊂ ( A n ) m ∩ Q and Q ∩ Z ( P λ ) = ∅ , then (7.4) implies N n,λ ≥ C ′ n ( ms − q ) for all λ ∈ U, n ∈ N . (7.5)Let Γ be the family of all tangent planes to Z ( P λ ) at x for λ ∈ U , x ∈ Q . Since s > qm − and R (Γ) satisfies the required transversality assumptions (by our choiceof Q and U ), we can apply Theorem 6.3 to each polynomial in the family e P λ,t,j ( x , . . . , x m ) = P λ ( x , . . . , x j − , t, x j +1 , . . . , x m ) ∈ P r,q,d ( m − , with λ ∈ U , t ∈ π j ( Q ), j ∈ [ m ]. This yields a constant C such that, letting R consist of the sets Z ( P λ ) ∩ Q ∩ { x j = t } , λ ∈ U, t ∈ [0 , d , j ∈ [ m ] , there is a positive probability that (7.4) holds together with the bound Z V ′ ν m − n ( x ) d H ( m − d − q ≤ C for all V ′ ∈ R , n ∈ N . (7.6)Indeed, by Theorem 6.3 (and compactness, making U smaller if needed), the prob-ability that (7.6) holds tends to 1 as C ↑ ∞ .Applying (7.1) in the proof of Theorem 7.1, we deduce the following: for any Q ′ ∈Q dn with centre t ′ , the number of cubes Q ⊂ A mn such that Q ∩ Z ( P λ ) ∩ Q ∩ { x j = t ′ } is bounded by C ′ n (( m − s − q ) . On the other hand, applying Lemma 6.1 as in theproof of Theorem 7.7, we see that if x ∈ Z ( P λ ) ∩ Q ∩ { x j = t } with t ∈ Q ′ , thenthere is x ′ ∈ B ( x, O (2 − n )) ∩ Z ( P λ ) ∩ Q ∩ { x ′ j = t ′ } where t ′ is again the centre of Q ′ (and the O constant is independent of λ, Q ′ ). Combining these facts, we deducethat if M n,λ,j,Q ′ denotes the number of cubes Q ⊂ A mn such that π j ( Q ) = Q ′ and Q ∩ Z ( P λ ) ∩ Q = ∅ , then M n,λ,j,Q ′ ≤ C ′ n (( m − s − q ) for all λ ∈ U, n ∈ N , j ∈ [ m ] and Q ′ ∈ Q dn . (7.7)To finish the proof, we will show that the claimed conclusion holds on the positiveprobability event that (7.5) and (7.7) hold. Suppose then that A ′ ⊂ A is a compactset such that ( A ′ ) m ∩ Z ( P λ ) ∩ Q = ∅ for some λ ∈ U . Since A ′ is compact, thesame still holds if we replace A ′ by the union of the cubes in Q dn hitting A ′ provided n is sufficiently large. By further enlarging A ′ slightly, we may assume without lossof generality that A ′ is the interior of a union of cubes Q dn for some n ∈ N .Fix n ≥ n . By (7.7), for each Q ′ ⊂ A n , Q ′ ∈ Q dn , there are at most C ′ n (( m − s − q ) cubes Q ∈ Q mdn with Q ⊂ A mn ∩ Q , Q ∩Z ( P λ ) = ∅ , and π j ( Q ) = Q ′ for some j ∈ [ m ]. ATTERNS IN RANDOM FRACTALS 63
Combining this with (7.5), it follows that A n \ A ′ contains at least C ′ n ( ms − q ) ( C ′ ) − − n (( m − s − q ) ≥ ε ns cubes in Q dn , where ε = C ′ /C ′ .Finally, the definition of ν as the weak limit of p − n L| A n implies (since we took A ′ to be open) that ν ( A \ A ′ ) ≥ ε , as desired. (cid:3) Given an open set U ⊂ Λ, we will say that the δ -Szemer´edi condition for U holdsif for any Borel A ′ ⊂ R d with ν ( A ′ ) > δν ( A ) there exist r > t ∈ R d such thatfor every λ ∈ U , Z ( P λ ) ∩ ( rA ′ + t ) m \ ∆ = ∅ . This is an event depending on the realization of the fractal percolation process.Since ν is a Radon measure, ‘Borel set’ may be replaced by ‘compact set’ withoutchanging the definition. Hence, Proposition 7.10 implies that for each λ ∈ Λ thereis a positive probability that the δ -Szemer´edi condition holds in a neighbourhood of λ , if δ is close enough to 1 (indeed, we can even take r = 1 , t = 0). In the next step,we upgrade “positive probability” to “full probability” at the price of changing thevalue of δ . Lemma 7.11.
If there is δ < such that P ( δ -Szemer´edi condition holds for U ) > , then there is δ ′ < such that the δ ′ -Szemer´edi condition for U holds almost surely.Proof. Denote η = P ( δ -Szemer´edi condition holds for U ) . Let X n be the number of those Q ⊂ A n , Q ∈ Q dn such that the fractal percolationmeasure ν Q induced on Q satisfies the δ -Szemer´edi condition for U . Fix ε > p ε < η/
8, where p ε = P ( k ν k < ε ). Denote by Z n the number of those Q ∈ Q dn with ν ( Q ) < ε − ns . We claim that almost surely the estimates X n ≥ η N n , Z n ≤ η N n and k ν k ≤ N n − ns hold for all large n . Indeed, recall from (3.1) that there is c > P ( N n ≤ c n ) < (1 − c ) n . (7.8)Conditioned on B n , Hoeffding’s inequality yields P (cid:16)(cid:16) X n ≥ η N n (cid:17) ∩ ( N n ≥ c n ) (cid:17) ≤ exp( − Ω η ( n )) , so Borel-Cantelli and Lemma (7.8) ensure that a.s. X n ≥ η N n for all large enough n . The rest of the claim follows in a similar way. Now suppose A ′ ⊂ A is a compact set such that ν ( A ′ ∩ Q ) ≤ δν ( Q ) for η N n cubes Q ∈ Q dn with Q ⊂ A n . Then at least η N n of these satisfy ν ( Q ) ≥ ε − ns , so that ν ( A \ A ′ ) ≥ (1 − δ ) η ε − ns N n ≥ (1 − δ ) ηε ν ( A ) . In other words, if ν ( A ′ ) > δ ′ ν ( A ), where δ ′ = 1 − (1 − δ ) ηε then ν ( A ′ ∩ Q ) > δν ( Q )for at least one Q ∈ Q dn such that ν Q satisfies the δ -Szemer´edi condition. Bydefinition, the δ -Szemer´edi condition for U is invariant under homothetic changes ofcoordinates. We thus conclude that the δ ′ -Szemer´edi condition for U holds almostsurely. (cid:3) Proof of Theorem 7.8.
By Proposition 7.10, for each λ ∈ Λ there exist a neighbour-hood U of λ and δ < P ( δ -Szemer´edi condition holds for U ) > . (By the regularity of ν , we may assume A ′ is compact.) By Lemma 7.11, thereexists δ ′ < P ( δ ′ -Szemer´edi condition holds for U ) = 1 . In particular, a.s. for all surviving cubes Q , the restricted process ν Q satisfies the δ ′ -Szemer´edi condition for U . Fix a realization such that this holds, and let A ′ ⊂ A be a measurable set with ν ( A ′ ) >
0. By a weak version of the Lebesgue densitypoint theorem, there is a cube Q (depending on A ′ ) such ν ( A ′ ∩ Q ) > δ ′ ν ( Q ). Hence,if Q = r [0 , d + t , then for each λ ∈ U , the set A ′ contains points x , . . . , x m suchthat P λ ( rx + t, . . . , rx m + t ) = 0. This proves the first claim in Theorem 7.8.The latter claim in Theorem 7.8 follows by covering the parameter space Λ bycountably many such sets U . (cid:3) References [1] Marc Carnovale. A relative Roth theorem in dense subsets of sparse pseudorandom fractals.Preprint, available at https://people.math.osu.edu/carnovale.2/Research/RFR.pdf ,2015. 3[2] Vincent Chan, Izabella Laba, and Malabika Pramanik. Finite configurations in sparse sets.
J.Anal. Math. , 128:289–335, 2016. 3, 54[3] Changhao Chen, Henna Koivusalo, Bing Li, and Ville Suomala. Projections of random cov-ering sets.
J. Fractal Geom. , 1(4):449–467, 2014. 16[4] D. Conlon and W. T. Gowers. Combinatorial theorems in sparse random sets.
Ann. of Math.(2) , 184(2):367–454, 2016. 4, 7[5] David Conlon, Jacob Fox, and Yufei Zhao. A relative Szemer´edi theorem.
Geom. Funct. Anal. ,25(3):733–762, 2015. 4[6] Roy O. Davies, J. M. Marstrand, and S. J. Taylor. On the intersections of transforms of linearsets.
Colloq. Math. , 7:237–243, 1959/1960. 2
ATTERNS IN RANDOM FRACTALS 65 [7] M. Burak Erdo˜gan. A bilinear Fourier extension theorem and applications to the distance setproblem.
Int. Math. Res. Not. , (23):1411–1425, 2005. 3[8] K. J. Falconer. On the Hausdorff dimensions of distance sets.
Mathematika , 32(2):206–212(1986), 1985. 3[9] K. J. Falconer. On a problem of Erd˝os on fractal combinatorial geometry.
J. Combin. TheorySer. A , 59(1):142–148, 1992. 3, 54[10] Kenneth Falconer.
Fractal geometry . John Wiley & Sons Inc., Hoboken, NJ, second edition,2003. Mathematical foundations and applications. 10[11] Kenneth Falconer and Xiong Jin. H¨older continuity of the liouville quantum gravity measure.Preprint, available at http://arxiv.org/1601.00556 , 2016. 16[12] Loukas Grafakos, Allan Greenleaf, Alex Iosevich, and Eyvindur Palsson. Multilinear gener-alized Radon transforms and point configurations.
Forum Math. , 27(4):2323–2360, 2015. 3,52[13] Ben Green and Terence Tao. The primes contain arbitrarily long arithmetic progressions.
Ann. of Math. (2) , 167(2):481–547, 2008. 2[14] Allan Greenleaf, Alex Iosevich, Bochen Liu, and Eyvindur Palsson. A group-theoretic view-point on Erd˝os-Falconer problems and the Mattila integral.
Rev. Mat. Iberoam. , 31(3):799–810, 2015. 3[15] Allan Greenleaf, Alex Iosevich, and Mihalis Mourgoglou. On volumes determined by subsetsof Euclidean space.
Forum Math. , 27(1):635–646, 2015. 52[16] Viktor Harangi, Tam´as Keleti, Gergely Kiss, P´eter Maga, Andr´as M´ath´e, Pertti Mattila, andBal´azs Strenner. How large dimension guarantees a given angle?
Monatsh. Math. , 171(2):169–187, 2013. 53[17] Kevin Henriot, Izabella Laba, and Malabika Pramanik. On polynomial configurations in frac-tal sets.
Anal. PDE , 9(5):1153–1184, 2016. 3[18] Alex Iosevich and Bochen Liu. Equilateral triangles in subsets of R d of large Hausdorff di-mension. Israel J. Math. to appear. Preprint available at http:/arxiv.org/1603.01907 . 3,54[19] Svante Janson. Large deviations for sums of partly dependent random variables.
RandomStructures Algorithms , 24(3):234–248, 2004. 19[20] Jean-Pierre Kahane. Positive martingales and random measures.
Chinese Ann. Math. Ser. B ,8(1):1–12, 1987. A Chinese summary appears in Chinese Ann. Math. Ser. A (1987), no. 1,136. 17[21] Tam´as Keleti. A 1-dimensional subset of the reals that intersects each of its translates in atmost a single point. Real Anal. Exchange , 24(2):843–844, 1998/99. 3[22] Steven G. Krantz and Harold R. Parks.
Geometric integration theory . Cornerstones.Birkh¨auser Boston, Inc., Boston, MA, 2008. 43, 46[23] Izabella Laba and Malabika Pramanik. Arithmetic progressions in sets of fractional dimension.
Geom. Funct. Anal. , 19(2):429–456, 2009. 3, 4[24] Quansheng Liu. Local dimensions of the branching measure on a Galton-Watson tree.
Ann.Inst. H. Poincar´e Probab. Statist. , 37(2):195–222, 2001. 13[25] Russell Lyons and Yuval Peres.
Probability on Trees and Networks . Cambridge Series in Sta-tistical and Probabilistic Mathematics. Cambridge University Press, 2017. 5, 13, 14, 15, 39[26] P´eter Maga. Full dimensional sets without given patterns.
Real Anal. Exchange , 36(1):79–90,2010/11. 3, 54 [27] Andr´as M´ath´e. Sets of large dimension not containing polynomial configurations.
Adv. Math. to appear. Preprint available at http://arxiv.org/1201.0548 ,. 53[28] Pertti Mattila.
Geometry of sets and measures in Euclidean spaces , volume 44 of
CambridgeStudies in Advanced Mathematics . Cambridge University Press, Cambridge, 1995. Fractalsand rectifiability. 39[29] Ursula Molter and Alexia Yavicoli. Small sets containing any pattern. Preprint, available at http://arxiv.org/1610.03804 , 2016. 2[30] Yuval Peres and Micha l Rams. Projections of the natural measure for percolation fractals.
Israel J. Math. , 214(2):539–552, 2016. 9[31] Micha l Rams and K´aroly Simon. The Dimension of Projections of Fractal Percolations.
J.Stat. Phys. , 154(3):633–655, 2014. 6, 51[32] K. F. Roth. On certain sets of integers.
J. London Math. Soc. , 28:104–109, 1953. 2[33] Mathias Schacht. Extremal results for random discrete structures.
Ann. of Math. (2) ,184(2):333–365, 2016. 4, 7[34] Pablo Shmerkin. Salem sets with no arithmetic progressions.
International Mathematics Re-search Notices . to appear. Published online https://doi.org/10.1093/imrn/rnr085 . 3[35] Pablo Shmerkin and Ville Suomala. A class of random measures, with applica-tions. In
Proceedings of the FARF III conference . to appear. Preprint available at https://arxiv.org/abs/1603.08156 . 8, 24, 28[36] Pablo Shmerkin and Ville Suomala. Spatially independent martingales, intersec-tions, and applications.
Mem. Amer. Math. Soc. to appear. Preprint available at http://arxiv.org/abs/1409.6707 . 4, 6, 8, 9, 17, 19, 22, 25, 33, 41, 57[37] Pablo Shmerkin and Ville Suomala. Sets which are not tube null and intersection propertiesof random measures.
J. Lond. Math. Soc. (2) , 91(2):405–422, 2015. 25[38] E. Szemer´edi. On sets of integers containing no k elements in arithmetic progression. ActaArith. , 27:199–245, 1975. Collection of articles in memory of Juri˘ı Vladimiroviˇc Linnik. 2[39] Thomas Wolff. Decay of circular means of Fourier transforms of measures.
Internat. Math.Res. Notices , (10):547–567, 1999. 3
Department of Mathematics and Statistics, Torcuato Di Tella University, andCONICET, Buenos Aires, Argentina
E-mail address : [email protected] URL : Department of Mathematical Sciences, University of Oulu, Finland
E-mail address : [email protected] URL ::