Distance sets, orthogonal projections, and passing to weak tangents
aa r X i v : . [ m a t h . M G ] M a r Distance sets, orthogonal projections,and passing to weak tangents
Jonathan M. FraserMarch 30, 2017
Abstract
We consider the Assouad dimension analogues of two important problems ingeometric measure theory. These problems are tied together by the common themeof ‘passing to weak tangents’. First, we solve the analogue of
Falconer’s distance setproblem for Assouad dimension in the plane: if a planar set has Assouad dimensiongreater than 1, then its distance set has Assouad dimension 1. We also obtainpartial results in higher dimensions. Second, we consider how Assouad dimensionbehaves under orthogonal projection . We extend the planar projection theorem ofFraser and Orponen to higher dimensions, provide estimates on the (Hausdorff)dimension of the exceptional set of projections, and provide a recipe for obtainingresults about restricted families of projections. We provide several illustrativeexamples throughout.
Mathematics Subject Classification
Key words and phrases : Assouad dimension, weak tangent, distance set, orthogonalprojection, exceptional set, restricted families.
The Assouad dimension is a fundamental notion in metric geometry, which goes backto Bouligand’s 1928 paper [B]. It also played a role in Furstenberg’s seminal workon galleries, which began in the 1960s, where it is referred to as the star dimension[Fu1, Fu2]. The notion rose to prominence again in the 1970s through the work ofAssouad which established powerful connections between the Assoaud dimension andembedding theory [A].We begin by recalling the definition, but refer the reader to [Fr, L, R] for more details.We consider subsets of d -dimensional Euclidean space R d ( d ∈ N ), but some of what wesay holds in more general spaces. For a non-empty bounded set E ⊂ R d and r > N r ( E ) be the smallest number of open sets with diameter less than or equal to r required to cover E . The Assouad dimension of a non-empty set F ⊆ R d is defined bydim A F = inf ( s > ∃ C >
0) ( ∀ R >
0) ( ∀ r ∈ (0 , R )) ( ∀ x ∈ F ) N r (cid:0) B ( x, R ) ∩ F (cid:1) C (cid:18) Rr (cid:19) s ) where B ( x, R ) denotes the closed ball centred at x with radius R . It is well-known thatthe Assouad dimension is always an upper bound for the Hausdorff dimension and (for1age 2 J. M. Fraserbounded sets) the upper box dimension. We write dim H for the Hausdorff dimension, H s for the s -dimensional Hausdorff (outer) measure, dim B for the upper box dimension,and dim P for the packing dimension. For precise definitions and basic properties of theseconcepts, we refer the reader to [F2].One of the most effective ways to bound the Assouad dimension of a set from belowis to use weak tangents ; an approach pioneered by Mackay and Tyson [MT]. Weaktangents are tools for capturing the extremal local structure of a set. Let K ( R d ) denotethe set of all non-empty compact subsets of R d . This is a complete metric space whenequipped with the Hausdorff metric d H defined by d H ( A, B ) = inf { δ > A ⊆ B δ and B ⊆ A δ } where, for any C ∈ K ( R d ), C δ = { x ∈ R d : | x − y | < δ for some y ∈ C } denotes the open δ -neighbourhood of C . We will also consider the space K ( X ) for afixed non-empty compact set X ⊆ R d . This is the set of all non-empty compact subsetsof X and, importantly, this is a compact subset of K ( R d ). Definition 1.1.
Let X ∈ K ( R d ) be a fixed reference set (usually the closed unit ball orsquare) and let E, F ⊆ R d be closed sets with E ⊆ X . Suppose there exists a sequenceof similarity maps T k : R d → R d such that d H ( E, T k ( F ) ∩ X ) → as k → ∞ . Then E is called a weak tangent to F . Recall that a similarity map T : R d → R d is a map of the form x cO ( x ) + t where c > similarity ratio ), O ∈ O ( R , d ) is a real orthogonal matrixand t ∈ R d is a translation. In most instances in this paper O will be the identity matrixand c will be large. The following important result of Mackay and Tyson will be usedthroughout the paper without being mentioned explicitly. Proposition 1.2. [MT, Proposition 6.1.5].
Let F ⊆ R d be closed, E ⊆ R d be compact,and suppose E is a weak tangent to F . Then dim A F > dim A E . It turns out that one can actually achieve the Assouad dimension as the Hausdorffdimension of a weak tangent. This beautiful fact, which is essentially due to Furstenberg,will be the key technical tool in this paper and will allow us to pass various geometricalproblems to weak tangents. This approach provides a mechanism for finding ‘Assouaddimension analogues’ of known results concerning the Hausdorff dimension (and otherdimensions) as well as proving stronger results, not known for other dimensions. Thuswe will obtain some fundamental results about the Assouad dimension via simple directarguments.
Theorem 1.3 (Furstenberg, K¨aenm¨aki-Ojala-Rossi) . Let F ⊆ R d be closed with dim A F = s ∈ (0 , d ] . Then there exists a compact set E ⊆ R d with H s ( E ) > which isa weak tangent to F . In particular, dim H E = dim A E = s . This result essentially appears in [Fu2, Section 5], without the conclusion that theweak tangent has positive H s measure and using different terminology: weak tangentsage 3 J. M. Fraserare replaced by ‘microsets’ and Assouad dimension by ‘star dimension’, dim ∗ . In fact theideas go back further to Furstenberg’s work in the 1960s, see [Fu1]. For an explicit ex-planation of the equivalence of Furstenberg’s setting and our setting, see [KR, Corollary5.2]. The theorem also appears using our terminology in [KOR, Propositions 5.7-5.8].K¨aenm¨aki-Ojala-Rossi give a direct and transparent proof that if d = 2, then there ex-ists a weak tangent E with dim H E = s and an inspection of their proof in fact yields H s ( E ) >
0. Moreover, their proof easily extends to the case of general d . The argumentpresented in the proof of [KOR, Propositions 5.7] is a development of an argument ofBishop-Peres [BP, Lemma 2.4.4] which showed that in the case d = 1 there exists a weaktangent with Hausdorff dimension equal to the upper box dimension of the original set.Also, [FY, Theorem 2.4] contains the above result in the case when s = d , i.e. for setswith full Assouad dimension. In this case one can actually find a weak tangent withnon-empty interior (which is stronger than positive H d measure).The term weak tangent is used in place of tangent for two reasons: there is no specialpoint in F at which we ‘zoom-in’, and the similarity ratios c associated with the maps T k need not ‘blow-up’. We reserve the word ‘tangent’ for weak tangents obtained by actuallyzooming in at a fixed point in the set. More precisely, we call E from the above definitiona tangent to F if the following additional assumptions are satisfied: X = B (0 ,
1) (theclosed unit ball); the sequence of similarity maps T k satisfies T k ( x ) = 0 for all k anda fixed point x ∈ F (i.e. we zoom-in to the point x ∈ F ); and c k ր ∞ as k → ∞ (i.e. we actually zoom-in). We end this section with the observation that weak tangentsare really needed to effectively study the Assouad dimension and that the more directnotion of tangent is not sufficient. Example 1.4.
There exists a compact set F ⊂ [0 , with dim A F = 1 , but such that alltangents to F have Assouad dimension equal to 0. We will provide the details of this example in Section 3.1.
Remark 1.5.
We recently learned that a similar example appears in [LR, Example 2.20].
Given a set F ⊆ R d , the distance set of F is defined by D ( F ) = {| x − y | : x, y ∈ F } . The distance set problem, which stems from the seminal paper of Falconer [F1], isto relate the dimensions of F with the dimensions of D ( F ). Falconer’s distance setconjecture refers to several related conjectures, one version of which is as follows:
Conjecture 2.1 (Falconer) . Let d > and F ⊆ R d be an analytic set. If dim H F > d/ ,then D ( F ) has positive Lebesgue measure (or even non-empty interior). In particular,it also has full Hausdorff dimension. age 4 J. M. FraserThere are numerous partial results in support of this conjecture, but the problemis still wide open, even in the plane. One may replace the Hausdorff dimension with adifferent notion of dimension, such as the packing, box counting or Assouad, and obtaina different conjecture. Conjecture 2.2.
Let dim denote the Hausdorff, packing, upper or lower box or Assouaddimension, let d > and F ⊆ R d be a bounded analytic set. If dim F > d/ , then dim D ( F ) = 1 . To the best of our knowledge, the different versions of this conjecture are all openfor all values of d >
2. See [Ma4, Conjecture 4.5] and the subsequent discussion relatedto the Hausdorff dimension version. An important recent result of Orponen [O2] isthat an AD-regular set with dimension at least 1 has a distance set with full packing dimension. Shmerkin later proved that these distance sets also have full (modified) lowerbox dimension [S]. For AD-regular sets all of the dimensions discussed here coincide andso these results do not resolve Conjecture 2.2 for any particular dimension. Our mainresult resolves Conjecture 2.2 for the Assouad dimension in the case d = 2. Note thatthe analyticity assumption is not needed. Theorem 2.3. If F ⊆ R is any set with dim A F > , then dim A D ( F ) = 1 . We will prove Theorem 2.3 in Section 3.2.1. Clearly, dim A F > D ( F ) has non-empty interior, but our result combined with [FY, Theorem 2.4]shows that it does guarantee the existence of a weak tangent to D ( F ) with non-emptyinterior. Moreover, in the setting of Conjecture 2.1 we obtain the following result on thelevel of weak tangents. Corollary 2.4. If F ⊆ R is any set with dim H F > , then D ( F ) has a weak tangentwith non-empty interior. In the higher dimensional case we obtain partial results. In particular, we showthat the Falconer-Erdo˘gan-Wolff bounds for Hausdorff dimension also hold for Assouaddimension (without any measureability assumptions), see [Ma4] for a recent survey of thestate of the art. It appears that the appropriate analogues of these bounds are not knownto hold for other notions of dimension, such as box-counting or packing dimension.
Theorem 2.5.
Let d > be an integer and F ⊆ R d be any set with dim A F d/ / .Then dim A D ( F ) > max (cid:26) A F + 2 − d , dim A F − d − (cid:27) . Moreover, if dim A F > d/ / , then dim A D ( F ) = 1 . We will prove Theorem 2.5 in Section 3.2.2. In fact our proof allows for a slightlystronger result, namely, that any bounds for the Hausdorff (or box) dimension versionof the distance set problem have direct Assouad dimension analogues.Using an elementary ‘product and project’ argument, it is easily seen (and well-known) that dim B D ( F ) B F for any bounded F ⊆ R d . The same bound holds forpacking dimension, dim P . However, there exists a compact set in R d (for any d ) withage 5 J. M. FraserHausdorff dimension 0, but whose distance set contains an interval, see [KNS, DMT]. Inparticular, [KNS, Proposition 3.1] provides a compact set A with Hausdorff dimension0 but for which A ∩ ( A + t ) = ∅ for all t ∈ [0 , d , for example. This proves that D ( A ) contains the interval [0 , √ d ]. Despite these examples, there is still some controlon the Hausdorff dimension of the distance sets of small sets in that one always hasdim H D ( F ) dim H F + dim P F . The situation for Assouad dimension is rather differentin that arbitrarily small sets from a dimension point of view can have distance sets withfull Assouad dimension. Example 2.6.
There exists a non-empty compact set F ⊆ [0 , with dim A F = 0 and dim A D ( F ) = 1 . We will provide the details of this example in Section 3.2.3. If one allows F to havepositive dimension, then the examples can be very simple and are also quite prevalent.In particular, for every s ∈ (0 , s butdim A D ( F ) = 1 . The following theorem readily yields such examples since self-similarsets satisfying the open set condition are AD-regular. Recall that a set is self-similar ifit is the unique non-empty compact set F satisfying F = [ i S i ( F )for a finite collection of similarity maps S i whose similarity ratios are all strictly lessthan 1. Such a collection of maps is called an iterated function system (IFS), and werefer the reader to [F2, Chapter 9] for more details, including the definition of the openset condition. Roughly speaking, the open set condition is satisfied if the ‘pieces’ S i ( F )do not overlap too much. Theorem 2.7.
Let F ⊆ R d be a self-similar set which is not a single point and supposethat two of the defining similarity ratios are given by a, b ∈ (0 , satisfying log a/ log b / ∈ Q . Then dim A D ( F ) = 1 . We will prove Theorem 2.7 in Section 3.2.3. The distance set problem for self-similarsets was considered by Orponen [O1]. He proved that if a self-similar set in the plane haspositive H measure, then the corresponding distance set has Hausdorff dimension one.Despite the previous examples, Theorem 2.3 is still sharp in the sense that one cannotguarantee that the distance set has full Assouad dimension for sets F with dim A F > s for any s < Example 2.8.
For every s ∈ [0 , , there exists a compact set F ⊆ [0 , with dim A F > s but dim A D ( F ) < . We will provide the details of this example in Section 3.2.4.age 6 J. M. Fraser
How dimension behaves under orthogonal projection is a classical problem in geomet-ric measure theory. It was first considered by Besicovitch and later by Marstrand; seeMarstrand’s seminal 1954 paper [M]. For integers k, d with 1 k < d , one considers pro-jections of R d onto k -dimensional subspaces. The k -dimensional subspaces of R d comewith a natural k ( d − k ) dimensional manifold structure, and so come equipped witha k ( d − k ) dimensional analogue of Lebesgue measure. We identify each k -dimensionalsubspace V with the orthogonal projection π : R d → V and denote the set of all such pro-jections as G d,k . We can thus make statements about almost all orthogonal projections π ∈ G d,k . The manifold G d,k is usually called the Grassmannian manifold and we referthe reader to [Ma2, Chapter 3] for more information on this manifold and its naturalmeasure. The classical result for Hausdorff dimension, often referred to as Marstrand’sprojection theorem, is that for a fixed Borel set F ⊆ R d , almost all π ∈ G d,k satisfydim H πF = min { k, dim H F } . This was first proved by Marstrand in the case d = 2 [M] and later by Mattila [Ma1] inthe general case. Similar results exist for upper and lower box dimension and packingdimension in the sense that for almost all π ∈ G d,k the dimension of πF is equal to aconstant. We refer the reader to the recent survey articles [FFJ, Ma3] for an overview ofthe rich and interesting theory of dimensions of projections. A striking difference in thecase of Assouad dimension, is that the dimension πF need not be almost surely constant.In particular, [FO, Theorem 2.5] provided an example of a set in the plane which projectsto sets with two different Assouad dimensions in positively many directions π ∈ G , .Our main result on projections shows that one can at least give an almost sure lowerbound on the Assouad dimension of πF . Theorem 2.9.
Let F ⊆ R d be any set and k < d be an integer. Then dim A πF > min { k, dim A F } for almost all π ∈ G d,k . We will prove this theorem in Section 3.3.1. The case when k = 1 and d = 2 wasproved by Fraser and Orponen, see [FO, Theorem 2.1], but our proof is completelydifferent. Recall that Fraser and Orponen proved that one cannot replace the almostsure lower bound with an almost sure equality. Our result is also sharp: for example, if F ⊆ R d is contained in a k -dimensional subspace, then it is easy to see that for almostall π ∈ G d,k we have dim A πF = dim A F .Interestingly, Theorem 2.9 is false if one replaces Assouad dimension by packing orupper box dimension. The packing/upper box dimension of πF is almost surely constant,but this constant can be strictly smaller than the minimum of k and the packing/upperbox dimension of F , see [FFJ, Ma3].We are also able to estimate the size of the exceptional set in the above theorem,i.e. the zero measure subset of G d,k where the lower bound from the theorem fails.In general, the exceptional set can be somewhere dense in G d,k and so considering theAssouad dimension of this set is the wrong approach (since the Assouad dimension ofage 7 J. M. Fraserany somewhere dense set is equal to the dimension of the ambient space). We thereforecompute the Hausdorff dimension of the exceptional set. Note that G d,k is a smoothmanifold of dimension k ( d − k ) and so our results are formulated to allow comparisonwith the dimension of the ambient space. Theorem 2.10.
Let F ⊆ R d be any set and k < d be an integer. For all < s min { k, dim A F } we have dim H { π ∈ G d,k : dim A πF < s } k ( d − k ) + s − min { k, dim A F } . We will prove this theorem in Section 3.3.2. Again, there does not appear to be adirect analogue of this exceptional set result for packing dimension or box dimension.See [O2] for some results in this direction and for an indication of why such analoguesare difficult (or even impossible) to obtain.Finally, we present some general results concerning restricted families of projections.Let P ⊆ R n be a Borel set with positive n -dimensional Lebesgue measure which pa-rameterises a family of projections { π t ∈ G d,k : t ∈ P } (we only assume that the map t π t is a bijection, but in practise it will usually have strong additional regularityproperties). One now wants to make statements about the dimensions of π t F for almostall t ∈ P in situations where { π t : t ∈ P } is a null set in G d,k , i.e. a genuinely restricted family of projections where Marstrand’s theorem yields no information directly. Thereare numerous results along these lines, most focusing on Hausdorff dimension, and werefer the reader to [FFJ, FO] for a survey of recent results. Rather than present severaldifferent Assouad dimension analogues, we give one ‘meta theorem’, which can apply inmany cases. Theorem 2.11.
Let F ⊆ R d be any set and k < d be an integer. Let P ⊆ R n be apositive measure set which parameterises a family of projections { π t ∈ G d,k : t ∈ P } asabove. Then for almost all t ∈ P we have dim A π t F > inf E ∈ K ( R d ):dim H E = dim A F essinf s ∈ P dim A π s E. We will prove this theorem in Section 3.3.3. This result gives a recipe for transformingresults on the Hausdorff dimension into results on the Assouad dimension. To motivatethis approach, we give one such example, which follows from a result of F¨assler andOrponen [FO] concerning projections of R onto lines in a ‘non-degenerate’ family ofdirections. Corollary 2.12.
Let F ⊆ R be any set and φ : (0 , → S be a C bijection such thatfor all t ∈ (0 , the vectors { φ ( t ) , φ ′ ( t ) , φ ′′ ( t ) } span R and let π t denote projection ontothe line in direction φ ( t ) .1. If dim A F = s > / , then there exists a constant σ ( s ) > / such that for almostall t ∈ (0 ,
1) dim A π t F > σ ( s ) .
2. If dim A F / , then for almost all t ∈ (0 , A π t F > dim A F. age 8 J. M. Fraser Proof.
This follows immediately from Theorem 2.11 and results from [FO]. For case ,F¨assler and Orponen proved that if an analytic set E ⊆ R has Hausdorff dimension s > /
2, then the packing dimension (and thus the Assouad dimension) of π t E is almostsurely bigger than a constant σ ( s ) > /
2, see [FO, Theorem 1.7]. For case , see [FO,Proposition 1.5]. Let F ⊆ [0 ,
1] be given by F = { } ∪ ∞ [ k =1 k [ l =0 n − k + l − k o . For each k let T k be the similarity defined by T k ( x ) = k − k ( x − − k ) and observe that T k ( F ) ∩ [0 ,
1] = k [ l =0 { l/k } → [0 , d H as k → ∞ . Therefore [0 ,
1] is a weak tangent to F and we may conclude thatdim A F = 1. Since F only has one accumulation point (at the origin) we only have toconsider tangents at this point. Indeed, any tangent to F at an isolated point is clearlya singleton and has Assouad dimension equal to zero. As such, suppose E is a tangentto F at 0. That is, there exists a sequence of similarity maps S n ( n >
1) of the form S n = c n x where 1 < c n ր ∞ as n → ∞ such that S n ( F ) ∩ B (0 , → E in d H as n → ∞ . Here we have assumed without loss of generality that the S n areorientation preserving, which we may do by considering a subsequence and introducinga reflection if necessary. Let F = { } ∪ ∞ [ k =1 n − k o and observe that dim A F = 0. We claim that E is also a tangent to F , which completesthe proof since any (weak) tangent to F necessarily has Assouad dimension 0. For each n let m ( n ) = min { k > c n − k } and note that m ( n ) → ∞ as n → ∞ . Observe that S n ( F ) ∩ B (0 , ⊆ { } ∪ ∞ [ k = m ( n ) k [ l =0 h c n − k , c n − k + c n k − k i age 9 J. M. Fraserand S n ( F ) ∩ B (0 ,
1) = { } ∪ ∞ [ k = m ( n ) n c n − k o and therefore d H (cid:16) S n ( F ) ∩ B (0 , , S n ( F ) ∩ B (0 , (cid:17) sup k > m ( n ) c n k − k = c n m ( n )4 − m ( n ) m ( n )2 − m ( n ) → . We conclude that S n ( F ) ∩ B (0 , → E in d H as n → ∞ , as required. The key technical lemma in proving our results on distance sets is the following. It statesthat one can pass questions on distance sets to weak tangents.
Lemma 3.1.
Let F ⊆ R d be a non-empty closed set and suppose E is a weak tangentto F . Then dim A D ( F ) > dim A D ( E ) . Proof.
Since E is a weak tangent to F , we may find a non-empty compact set X ⊆ R d and a sequence of similarity maps T k : R d → R d such that T k ( F ) ∩ X → E (3.1)in d H as k → ∞ . We may clearly assume that X is not a single point and we write | X | ∈ (0 , ∞ ) for the diameter of X . Also, for each k , write c k ∈ (0 , ∞ ) for the similarityratio of T k . Consider the sequence of compact sets given by c k D ( F ) ∩ [0 , | X | ]where c k D ( F ) = { c k z : z ∈ D ( F ) } and take a strictly increasing infinite sequence ofintegers ( k n ) n> such that c k n D ( F ) ∩ [0 , | X | ] converges in d H to a compact set B . Wemay do this since ( K ([0 , | X | ]) , d H ) is compact. In particular, B is a weak tangent to D ( F ) and so dim A D ( F ) > dim A B. Thus to complete the proof it suffices to show that D ( E ) ⊆ B . Let z = | x − y | ∈ D ( E )for some x, y ∈ E . It follows from (3.1) that we can find a sequence of pairs x k , y k ∈ T k ( F ) ∩ X such that x k → x and y k → y . For each k we have T − k ( x k ) , T − k ( y k ) ∈ F andso c − k | x k − y k | = | T − k ( x k ) − T − k ( y k ) | ∈ D ( F ) . Moreover, | x k − y k | | X | and therefore | x k − y k | ∈ c k D ( F ) ∩ [0 , | X | ] . age 10 J. M. FraserIt follows that z = | x − y | = lim k →∞ | x k − y k | = lim n →∞ | x k n − y k n | ∈ B which completes the proof.We are now ready to prove Theorem 2.3. Let F ⊆ R be a closed set with dim A F = s >
1. We will deal with the non-closed case at the end of the proof.It follows from Theorem 1.3 that F has a weak tangent E such that H s ( E ) > E ′ ⊆ E with positive and finite H s measure, see [F2,Theorem 4.10], and define ν = 1 H s ( E ′ ) H s | E ′ to be the normalised restriction of H s to E ′ . Thus ν is a Borel probability measuresupported on a compact set E ′ of positive H s measure. Without loss of generality wemay assume that E ′ is contained in [0 , and note that the ν measure of the boundaryof [0 , (or any square) is zero. We will now employ the theory of CP-chains, whichwere introduced by Furstenberg in the seminal paper [Fu2] building on his earlier workfrom the 1960s, see [Fu1]. The theory has recently been developed by Hochman [H] andHochman-Shmerkin [HS] and has proved a powerful tool in studying many geometricproblems. The idea is to apply ideas from ergodic theory to the measure valued processgenerated by zooming in at a point in the support of a fractal measure.Let M denote the collection of all Borel probability measure supported on [0 , ,endowed with the topology of weak convergence. Let E be the collection of all half opendyadic boxes contained in [0 , oriented with the coordinate axes. By half open dyadicbox we mean a set of the form [ a, b ) × [ c, d ) where both [ a, b ) and [ c, d ) are dyadic intervalsof the same length. For x ∈ [0 , , write ∆ k ( x ) to denote the unique k th generation boxin E containing x . For B ∈ E , let T B : R → R be the unique rotation and reflectionfree similarity that maps B onto [0 , d . For µ ∈ M and B ∈ E such that µ ( B ) >
0, wewrite µ B = 1 µ ( B ) µ | B ◦ T − B ∈ M . The measures µ ∆ k ( x ) are called (dyadic) minimeasures (at x ) and weak limits of se-quences of minimeasures where the level k → ∞ are called (dyadic) micromeasures (at x ). We denote the set of all micromeasures of µ by Micro( µ ). A CP-chain is a stationaryMarkov process ( µ n , x n ) ∞ n =1 on the state space { ( µ, x ) : µ ∈ M , x ∈ [0 , , and for all k ∈ N , µ (∆ k ( x )) > } where the transition probability is given by( µ, x ) → (cid:16) µ ∆ ( x ) , T ∆ ( x ) ( x ) (cid:17) with probability µ (∆ ( x )). There is a minor technical issue here if µ gives positivemeasure to the boundary of the dyadic boxes in E , but we can omit discussion of thissince we will apply the theory to ν which does not have this property. For convenienceage 11 J. M. Fraserwe assume from now on that µ gives zero measure to the boundary of all dyadic boxes.The measure component of the stationary distribution for the process described aboveis a measure Q supported on M and is called a CP-distribution . A CP-chain is said tobe ergodic if Q is ergodic. We say a measure µ ∈ M generates a CP-chain with measurecomponent Q if at µ almost every x ∈ [0 , , the scenery distributions1 N N X k =1 δ µ ∆ k ( x ) converge weakly to Q as N → ∞ and for every q ∈ N , there exists a distribution Q q on M such that at µ almost every x ∈ [0 , , the q -sparse scenery distributions1 N N X k =1 δ µ ∆ qk ( x ) converge weakly to Q q as N → ∞ . Here the distributions Q and Q q are necessarilysupported on the micromeasures Micro( µ ). We refer the reader to [HS, Section 7] formore details on CP-chains. They have proved to be of central importance in severalproblems on geometric measure theory in the last few years and, in particular, measureswhich generate ergodic CP-chains have many useful properties. We will use the follow-ing result of Ferguson-Fraser-Sahlsten [FFS, Theorem 1.7] which relates CP-chains todistance sets. Recall that the (lower) Hausdorff dimension of a measure is defined bydim H µ = inf { dim H F : µ ( F ) > } and note that dim H ν = s . Theorem 3.2 (Ferguson-Fraser-Sahlsten) . Let µ ∈ M be a measure which generates anergodic CP-chain and is supported on a set X of positive length, i.e. H ( X ) > . Then dim H D ( X ) > min { , dim H µ } . The dimension of a CP-chain is the average of the dimensions of micromeasures withrespect to the measure component of the chain, i.e. Z dim H ν dQ ( ν ) , but for an ergodic CP-chain the micromeasures are almost surely exact dimensional witha common ‘exact dimension’, [HS, Lemma 7.9]. Recall that a measure µ is called exactdimensional if the local dimension lim r → log µ ( B ( x, r ))log r exists and equals some constant α at almost every point in the support. In this case, wealso have dim H µ = α .Hochman and Shmerkin proved that for any µ ∈ M , there exists an ergodic CP-chain whose measure component Q is supported on Micro( µ ) and has dimension atleast dim H µ , see [HS, Theorem 7.10]. Moreover, [HS, Theorem 7.7] tells us that Q -almost all micromeasures generate this CP-chain. In particular, for ν defined aboveage 12 J. M. Fraserwe can guarantee the existence of a micromeasure ν ′ ∈ Micro( ν ) which generates anergodic CP-chain of dimension at least s and satisfies dim H ν ′ > s >
1. This guaranteesthat the support of ν ′ has positive length. Putting these facts together, there exists amicromeasure ν ′ of ν which generates an ergodic CP-chain and is supported on a set X of positive length. We now require the following simple general lemma. Lemma 3.3.
Let Z ⊆ R d be a fixed compact set and µ k be a sequence of Borel probabilitymeasures with supports denoted by Y k ⊆ Z such that µ k weakly converges to µ and Y k converges to Y in the Hausdorff metric. Then the support of µ is a subset of Y .Proof. For
A, B ∈ K ( Z ), let ρ H ( A, B ) = inf { δ > A ⊆ B δ } . Write supp( µ ) for the support of µ . We claim that ρ H (supp( µ ) , Y k ) → k → ∞ .Suppose not, in which case there exists ε > x ∈ supp( µ ) such that there existarbitrarily large k such that Y k ∩ B ( x, ε ) = ∅ where B ( x, ε ) denotes the open ball centred at x with radius ε . Therefore µ ( B ( x, ε )) lim inf k →∞ µ k ( B ( x, ε )) = 0which is a contradiction. We conclude that ρ H (supp( µ ) , Y ) ρ H (supp( µ ) , Y k ) + d H ( Y k , Y ) → µ is equal to Y . The sequence µ k = (1 /k ) H | [0 , + (1 − /k ) δ provides a counter example, where δ denotes a point mass at 0.Since ν ′ is a micromeasure of ν , it is the weak limit of a sequence of minimeasures ν B k (where B k is a dyadic square) which are supported on T B k ( E ′ ) ∩ [0 , ∈ K ([0 , ).Since K ([0 , ) is compact we can assume the sequence T B k ( E ′ ) ∩ [0 , converges in d H to a non-empty compact set E ′′ ⊆ [0 , , which is therefore a weak tangent to E ′ . Itfollows from Lemma 3.3 that X = supp( ν ′ ) ⊆ E ′′ .The desired result now follows by piecing together the above:dim A D ( F ) > dim A D ( E ) by Lemma 3.1 since E is a weak tangent to F > dim A D ( E ′ ) since E ′ ⊆ E > dim A D ( E ′′ ) by Lemma 3.1 since E ′′ is a weak tangent to E ′ > dim A D ( X ) since X ⊆ E ′′ > dim H D ( X ) > min { , dim H ν ′ } by Theorem 3.2= 1which completes the proof.age 13 J. M. FraserAll that remains is the case when F is not closed. However, dim A F = dim A F > D ( F ) ⊇ D ( F ). Therefore, using the result in the closed case, we havedim A D ( F ) = dim A D ( F ) > dim A D ( F ) = 1as required. This theorem follows immediately by combining Theorem 1.3, Lemma 3.1 and the resultsof Falconer and Erdo˘gan on Hausdorff dimension. Let F ⊆ R d be any set. UsingTheorem 1.3 we can find a compact (and so Borel) set E ⊆ R d which is a weak tangentto F with dim H E = dim A F = dim A F . Combining the well-known results of Falconerand Erdo˘gan, see [Ma4, Chapter 15], we havedim H D ( E ) > max (cid:26) A F + 2 − d , dim A F − d − (cid:27) provided dim H E d/ / H D ( E ) = 1 otherwise. The desired result thenfollows from Lemma 3.1 sincedim A D ( F ) = dim A D ( F ) > dim A D ( F ) > dim A D ( E ) > dim H D ( E ) . We begin by proving Theorem 2.7 and then we will adapt the construction of a self-similar set to satisfy the conditions of Example 2.6. Let F ⊂ R d be a self-similar setwhich is not a single point. Let x, y ∈ F be distinct points and let ∆ := | x − y | >
0. Let S a be one of the defining similarity maps with contraction ratio a and S b be one of thedefining similarity maps with contraction ratio b . For all integers m, n > S ma ◦ S nb ( x ) , S ma ◦ S nb ( y ) ∈ F and therefore | S ma ◦ S nb ( x ) − S ma ◦ S nb ( y ) | = a m b n ∆ ∈ D ( F ) . The proof of Theorem 2.7 is now similar to the example in [Fr, Section 3.1], but weinclude the details for completeness. We will show that [0 ,
1] is a weak tangent to D ( F )which proves the theorem. For each integer k > T k : [0 , → [0 ,
1] be defined by T k ( x ) = ∆ − b − k x and, using compactness of K ([0 , T k ( D ( F )) ∩ [0 ,
1] in the Hausdorff metric, the limit of which is a weak tangent to D ( F ). Since (cid:8) a m b n : m, n ∈ Z , m > , n > − k (cid:9) ∩ [0 , ⊂ T k ( D ( F )) ∩ [0 , k > { a m b n : m ∈ N , n ∈ Z } ∩ [0 , . age 14 J. M. FraserHowever, it follows almost immediately from the assumption on a and b that this set issimply [0 , { m log a + n log b : m ∈ N , n ∈ Z } is dense in ( −∞ , m log a + n log b = n log a (cid:18) mn + log b log a (cid:19) and by Dirichlet’s theorem on Diophantine approximation combined with the irrational-ity of log b/ log a we can find infinitely many integers n > < (cid:12)(cid:12)(cid:12) mn + log b log a (cid:12)(cid:12)(cid:12) < /n for some integer m . Therefore for any ε > m, n such that0 < | m log a + n log b | < ε and by scaling m, n by each positive integer in turn we can find infinite (one sided)arithmetic progressions with arbitrarily small gap length inside { m log a + n log b : m ∈ N , n ∈ Z } which completes the proof.We will now show how to build an example with Assouad dimension 0 since theexamples provided by Theorem 2.7 all have strictly positive Assouad dimension. Thisis the content of Example 2.6. Fix a, b ∈ (0 ,
1) with log a/ log b / ∈ Q as above. For eachinteger k >
1, consider the IFS I ( k ) consisting of the two maps S k : x a k x and S k : x b k x + (1 − b k )and let N ( k ) be a large positive integer which we will specify later. Let θ = ( θ , θ , . . . ) ∈ N N be the infinite integer sequence defined by beginning with N (1) 1s, and followingwith N (2) 2s, and so on. In other words, θ is the unique word with non-decreasingentries such that for all integers k there are precisely N ( k ) occurrences of the integer k .Also let Φ k : K ([0 , → K ([0 , I ( k ), i.e.Φ k ( X ) = S k ( X ) ∪ S k ( X ) . Finally, let E = ∞ \ n =1 Φ θ n ◦ · · · ◦ Φ θ ([0 , E is a non-empty compact subset of [0 , k > E into finitely many pieces, each of which is a subset of the attractorof I ( k ). Since the Assouad dimension is finitely stable, this means that we can boundthe Assouad dimension of E by the similarity dimension of the attractor of I ( k ) for all k , see [F2, Chapter 9] and [Fr]. In particular, the similarity dimension associated with I ( k ) is the unique real solution s ( k ) of a s ( k )2 k + b s ( k )2 k = 1and so dim A E inf k > s ( k ) = 0 . age 15 J. M. FraserAll remains is to prove that we may choose the integers N ( k ) such that the key featureof the systems we use is preserved, i.e. dim A D ( E ) = 1.By construction, for all k > D ( E ) ⊇ k − Y l =1 a ( l − N ( l − ! n a k m b k n : m, n ∈ Z , m, n N ( k ) o . These points are found similar to above, but by looking at the left most interval at level N (1) + · · · + N ( k −
1) in the construction, which has length k − Y l =1 a ( l − N ( l − ! , and then looking at end points of intervals within this interval for the next N ( k ) levels.For each k >
1, let T k be defined by T k ( x ) = k − Y l =1 a ( l − N ( l − ! − b − k N ( k ) / ( x )and assume for convenience that N ( k ) is even. It follows that T k ( D ( E )) ∩ [0 , ⊇ n a k m b k n : m, n ∈ Z , m N ( k ) , − N ( k ) / n N ( k ) / o ∩ [0 , . Now choose N ( k ) sufficiently large (and even) such that the Hausdorff distance betweenthe set above and the closure of the set I := n a k m b k n : m, n ∈ Z , m < ∞ , −∞ < n < ∞ o ∩ [0 , /k . However, since log a k log b k = log a log b / ∈ Q , we have already seen that the closure of I is simply the unit interval [0 , T k ( D ( E )) ∩ [0 , → [0 ,
1] in the Hausdorff distance as k → ∞ . In particular, [0 , D ( E ) and we conclude that dim A D ( E ) = 1 as required. Let s ∈ [0 ,
1) and let N > N + 1) / − N s > K be aninteger satisfying N s K < ( N + 1) /
2. For all i ∈ { , . . . , K − } let S i : [0 , → [0 , S i ( x ) = ( x + 2 i ) /N age 16 J. M. Fraserand F ⊆ [0 ,
1] be the self-similar set associated with the IFS { S i } K − i =0 . Since the definingIFS satisfies the open set condition, it follows thatdim A F = dim H F = log K log N > s. Consider the set π ( F × F ) where π is orthogonal projection onto the subspace of R spanned by (1 , −
1) identified with R . This is a self-similar set defined by 2 K − /N . Moreover, by our choice ofindexing, a simple geometric argument shows that the open set condition is satisfied forthis system. The map φ defined by φ ( x ) = | x | is bi-Lipschitz on ( −∞ ,
0) and [0 , ∞ ) andso cannot increase Assouad dimension (the fact that it is Lipschitz on the whole of R is not enough to guarantee this, see [L, Example A.6 2] or [Fr, Section 3.1]). Moreover, φ ( π ( F × F )) = D ( F ) and thereforedim A D ( F ) dim A π ( F × F ) log 2 K − N <
Let F ⊆ R d be any set, and let k ∈ [1 , d ) be an integer. Since for any π ∈ G d,k we have π ( F ) ⊆ π ( F ) and the Assouad dimension is stable under taking closure, we may assumethat F is closed to begin with. It follows from Theorem 1.3 that there exists a compactset E ⊆ R d which is a weak tangent to F such that dim H E = dim A F . In particular,there exists a sequence of similarity maps T k on R d and a compact set X ⊆ R d such that T k ( F ) ∩ X → E (3.2)in d H as k → ∞ . Moreover, we may assume that the T k are homothetic, i.e. of the form T k ( x ) = c k x + t k for a real constant c k > t k ∈ R d . In fact the proof ofTheorem 1.3 given in [KOR] yields homothetic maps directly, but it is also easy to provefrom the statement of Theorem 1.3 in this paper. Suppose T k ( x ) = c k O k x + t k where O k ∈ O ( R , d ) is a not necessarily trivial orthogonal component and O ( R , d ) is the realorthogonal group. Since O ( R , d ) is compact in the topology of uniform convergence, wemay assume that O k → O uniformly for some fixed O ∈ O ( R , d ) by taking a subsequenceif necessary. Using continuity of O − and (3.2) it is easily verified that( c k ( F ) + O − ( t k )) ∩ O − ( X ) → O − ( E )and therefore O − ( E ) is a weak tangent to F with all the desired properties.In what follows it is convenient to identify π ( R d ) with R k in the natural way. Definea map π ◦ T k ◦ π − from π ( R d ) to itself by { π ◦ T k ◦ π − ( x ) } = { π ( T k ( y )) : π ( y ) = x } age 17 J. M. Fraserand observe that since T k is assumed to be homothetic this is well-defined, i.e. the set { π ( T k ( y )) : π ( y ) = x } is a singleton. Moreover, writing T k ( x ) = c k x + t k , we have for x ∈ π ( R d ) that π ◦ T k ◦ π − ( x ) = π ( c k π − ( x ) + t k ) = c k x + π ( t k )and so π ◦ T k ◦ π − is itself a (homothetic) similarity. Since π : K ( R d ) → K ( R k ) iscontinuous, it follows from (3.2) that (cid:0) π ◦ T k ◦ π − (cid:1) ( π ( F )) ∩ π ( X ) = π ( T k ( F )) ∩ π ( X ) ⊇ π ( T k F ∩ X ) → π ( E ) (3.3)in d H as k → ∞ . Note that π ( X ) is a compact subset of π ( R d ) and K ( π ( X )) is compactand so we may assume, by taking a subsequence if necessary, that ( π ◦ T k ◦ π − )( π ( F )) ∩ π ( X ) converges to a compact set E ′ ⊆ π ( X ) in d H as k → ∞ . In particular, E ′ is aweak tangent to π ( F ) and it follows from (3.3) that E ′ ⊇ π ( E ).Theorem 2.9 now follows immediately. We demonstrated above that for all π ∈ G d,k ,the set π ( E ) is a subset of a weak tangent to π ( F ). It therefore follows from Marstrand’sclassical projection theorem for Hausdorff dimension that for almost all π ∈ G d,k we havedim A π ( F ) > dim A π ( E ) > dim H π ( E ) = min { k, dim H E } = min { k, dim A F } as required. Theorem 2.10 follows by combining the argument of the previous section with knownestimates for the Hausdorff dimension of the set of exceptions to Marstrand’s classicalprojection theorem for Hausdorff dimension. In particular, let E be as before and recallthat dim H E = dim A F and for all π ∈ G d,k we have dim A π ( F ) > dim H π ( E ). Therefore,for 0 < s min { k, dim A F } = min { k, dim H E } , we have { π ∈ G d,k : dim A π ( F ) < s } ⊆ { π ∈ G d,k : dim H π ( E ) < s } and so applying the known bounds, which can be found in [Ma4, Corollary 5.12] forexample, yielddim H { π ∈ G d,k : dim A π ( F ) < s } dim H { π ∈ G d,k : dim H π ( E ) < s } k ( d − k ) − (min { k, dim H E } − s )= k ( d − k ) − (min { k, dim A F } − s )as required. This theorem is proved in a similar way to Theorem 2.9. Let E be as before and recallthat dim H E = dim A F and for all t ∈ P we have dim A π t ( F ) > dim A π t ( E ). It followsthat for almost all t ∈ P we havedim A π t ( F ) > dim A π t ( E ) > essinf s ∈ P dim A π s E > inf E ∈ K ( R d ):dim H E = dim A F essinf s ∈ P dim A π s E age 18 J. M. Fraserwhich completes the proof. Acknowledgements
The author is supported by a
Leverhulme Trust Research Fellowship (RF-2016-500). Hethanks Antti K¨aenm¨aki, John Mackay, and Pablo Shmerkin for helpful discussions.
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