A nonlinear projection theorem for Assouad dimension and applications
aa r X i v : . [ m a t h . M G ] A p r A nonlinear projection theorem for Assouad dimensionand applications
Jonathan M. FraserApril 28, 2020
Abstract
We prove a general nonlinear projection theorem for Assouad dimension. Thistheorem has several applications including to distance sets, radial projections, andsum-product phenomena. In the setting of distance sets we are able to completelyresolve the planar distance set problem for Assouad dimension, both dealing with theawkward ‘critical case’ and providing sharp estimates for sets with Assouad dimensionless than 1. In the higher dimensional setting we connect the problem to the dimensionof the set of exceptions in a related (orthogonal) projection theorem. We also obtainresults on pinned distance sets and our results still hold when the distances are takenwith respect to a sufficiently curved norm. As another application we prove a radialprojection theorem for Assouad dimension with sharp estimates on the Hausdorffdimension of the exceptional set.
Mathematics Subject Classification
Key words and phrases : Assouad dimension, nonlinear projections, distance sets,radial projections, exceptional set, Hausdorff dimension, sum-product theorem.
How dimension behaves under projection is a well-studied and important problem in geo-metric measure theory with many varied applications. The classical setting is to relate theHausdorff dimension of a set F ⊆ R n with the Hausdorff dimension of π V ( F ) for generic V ∈ G ( n, m ). Here and throughout G ( n, m ) denotes the Grassmannian manifold consist-ing of m -dimensional subspaces of R n and π V denotes orthogonal projection from R n to V ∈ G ( n, m ). We write dim H E for the Hausdorff dimension of a set E . The seminalMarstrand-Mattila projection theorem states that for Borel sets F ⊆ R n dim H π V ( F ) = min { dim H F, m } (1.1)for almost all V ∈ G ( n, m ). Here ‘almost all’ is with respect to the Grassmannian measure,which is the appropriate analogue of m ( n − m )-dimensional Lebesgue measure on G ( n, m ).The planar case of this result goes back to Marstrand’s 1954 paper [M54] and the generalcase was proved by Mattila [M75]. This result has inspired much work in geometric measuretheory, fractal geometry, harmonic analysis, ergodic theory and many other areas.This paper is concerned with the Assouad dimension, which is a well-studied notion ofdimension with key applications in embedding theory, quasi-conformal geometry and fractalgeometry. The analogue of the Marstrand-Mattila projection theorem for Assouad dimen-sion was proved in [F18, Theorem 2.9], the planar case having been previously established1age 2 J. M. Fraserby Fraser and Orponen [FO17]. We write dim A E for the Assouad dimension of a set E .The result is that for any non-empty set F ⊆ R n dim A π V ( F ) > min { dim A F, m } (1.2)for almost all V ∈ G ( n, m ). An interesting feature of this result is that the inequality cannotbe replaced by equality in general. This latter fact was proved in [FO17] and in [FK] it wasproved that, apart from satisfying (1.2) almost surely, the behaviour of dim A π V ( F ) can bevery wild. Our projection theorems will share this phenomenon and we make no furthermention of this.We are concerned with parameterised families of nonlinear projections, rather thanthe orthogonal projections π V . Our treatment and exposition takes some inspiration fromthe nonlinear projection theorems of Peres and Schlag [PS00], which are primarily in thesetting of Hausdorff dimension of sets and measures. The work of Peres and Schlag hasproved influential, with the concept of transversality at the centre. Their general nonlinearprojection theorems have applications in several areas including radial projections, distancesets, Bernoulli convolutions, sumsets, and many other ‘nonlinear’ problems. Our mainresult, Theorem 2.2, is a general nonlinear projection theorem for Assouad dimension, andthis too has many applications. Most strikingly to distance sets, where we are able tocompletely resolve the planar distance set problem for Assouad dimension, see Theorem3.1. Specifically, we prove that the Assouad dimension of the distance set of a set F in theplane is at least min { dim A F, } . In the higher dimensional setting we connect the problemto the dimension of the set of exceptions to (1.2), see Theorem 3.3. We also obtain resultsfor pinned distance sets and for distance sets where the distances are taken with respect to a‘sufficiently curved’ norm. Our proofs use tools from geometric measure theory, such as thetheory of weak tangents [MT10, KOR18]; fractal geometry, such as Orponen’s projectiontheorem for Assouad dimension [O] and transversality; and also differential geometry, withlinearisation the underlying principle.For background on fractal geometry, including Hausdorff dimension and the dimensiontheory of projections, see the books [F14a, M95] and the recent survey articles on projections[FFJ15, M14]. For background on the Assouad dimension, see the books [F20, R11], and forrecent results on the Assouad dimension of orthogonal projections, see [F18, FK, FO17, O].There has recently been intensive interest in nonlinear projections in a variety of contexts.For example, see [B17, BLZ16, HS12, S].For concreteness we recall the definition of the Assouad dimension, although we willnot use the definition directly. Given F ⊆ R n , the Assouad dimension of F is defined tobe the infimum of α > C > x ∈ F andscales 0 < r < R , the intersection of F with the ball B ( x, R ) may be covered by fewer than C ( R/r ) α sets of diameter r . In particular, 0 dim H F dim A F n . Our main result is a general nonlinear projection theorem for Assouad dimension. Thenonlinear projections we consider are defined in Definition 2.1. The definition may seemtechnical, but in the applications which follow it will be obvious that these conditions aresatisfied.age 3 J. M. Fraser
Definition 2.1.
We call ( { Π t : t ∈ Ω } , µ, P ) a generalised family of projections of R n ofrank m > if Ω is a metric space, µ a Borel measure on Ω , P a Borel measure on G ( n, m ) and:1. (Domain) For all t ∈ Ω , Π t is a function mapping R n into itself.2. (Differentiability) For all z ∈ R n , Π t is a C map of constant rank m in some openneighbourhood of z for µ almost all t ∈ Ω . That is, for µ almost all t , Π t is con-tinuously differentiable in a neighbourhood of z and the Jacobian J z ′ Π t is a rank m matrix for all z ′ sufficiently close to z .In particular, this means that for all z ∈ R n the map T z : Ω → G ( n, m ) given by T z ( t ) = ker( J z Π t ) ⊥ is well-defined almost everywhere (using the rank nullity theorem).3. (Absolute continuity) For all z ∈ R n , µ ◦ T − z ≪ P . Note that the Jacobian derivatives J z Π t appearing in Definition 2.1 need not be pro-jection matrices. In most applications, for all z , Π t will be smooth in a neighbourhood of z for all but at most one point t ∈ Ω. One can think of the absolute continuity assumptionin terms of transversality of the family { Π t } t . Theorem 2.2.
Let ( { Π t : t ∈ Ω } , µ, P ) denote a generalised family of projections of R n ofrank m > . For all non-empty bounded F ⊆ R n , dim A Π t ( F ) > inf E ⊆ R n :dim H E =dim A F essinf V ∼ P dim A π V ( E ) for µ almost all t ∈ Ω . We chose to use general Borel measures P on G ( n, m ) rather than the usual Grass-mannian measure because this allows us to deduce dimension estimates for the exceptionalset. However, the most direct application of Theorem 2.2 is when P is the Grassmannianmeasure. Corollary 2.3.
Let ( { Π t : t ∈ Ω } , µ, P ) denote a generalised family of projections of R n ofrank m > , where P is the Grassmannian measure. For all non-empty bounded F ⊆ R n , dim A Π t ( F ) > min { dim A F, m } for µ almost all t ∈ Ω .Proof. This follows from Theorem 2.2 and (1.2).It is also of interest to investigate the exceptional set in Corollary 2.3. Theorem 2.2also allows one to obtain estimates on the Hausdorff dimension of the exceptional set byrelating it to the Hausdorff dimension of the exceptional set in the setting of orthogonalprojections. We write H s for the s -dimensional Hausdorff (outer) measure.age 4 J. M. Fraser Corollary 2.4.
Suppose ( { Π t : t ∈ Ω } , H s , H u ) is a generalised family of projections of R n of rank m > for all u > sup dim H { V ∈ G ( n, m ) : dim A π V ( E ) < λ } where the supremum is taken over all non-empty E ⊆ R n with dim H E = dim A F . For allnon-empty bounded F ⊆ R n , dim A Π t ( F ) > λ for all t ∈ Ω outside of a set of exceptions of Hausdorff dimension at most s .Proof. This follows from Theorem 2.2 since, for all E ⊆ R n with dim H E = dim A F ,essinf V ∼H u dim A π V ( E ) > λ. When applying Corollary 2.4 it is useful to be able to estimate θ ( s, n, m ) := sup dim H { V ∈ G ( n, m ) : dim A π V F < min { dim A F, m }} (2.1)where the supremum is taken over all sets F ⊆ R n with dim A F = s . It was proved in [F18]that, for all integers n > m > s ∈ [0 , n ], θ ( s, n, m ) m ( n − m ) − | m − s | . (2.2)These bounds are simply the known (sharp) bounds for the set of exceptions to (1.1)translated to the Assouad dimension setting (1.2). Corollary 2.4 is especially useful when n = 2 and m = 1 since Orponen’s projection theorem [O] provides the sharp estimate onthe Hausdorff dimension of the set of exceptions to (1.2) in the planar case. Corollary 2.5.
Suppose ( { Π t : t ∈ Ω } , H s , H u ) is a generalised family of projections of R of rank for all u > . For all non-empty bounded F ⊆ R , dim A Π t ( F ) > min { dim A F, } for all t ∈ Ω outside of a set of exceptions of Hausdorff dimension at most s .Proof. This follows from Corollary 2.4 and Orponen’s projection theorem [O, Theorem 1.1],which shows thatdim H { V ∈ G (2 ,
1) : dim A π V ( E ) < min { dim A E, }} = 0for all non-empty E ⊆ R . In particular, θ ( s, ,
1) = 0 for all s ∈ [0 , F contained in asubset U ⊆ R n . In this case the results in this section can be applied under the weakerassumption that the domain of each Π t is an open set U ⊇ U , and the differentiability andabsolute continuity assumptions hold only for all z ∈ U . This version of the theorem canbe deduced directly from Theorem 2.2 appealing to the Whitney extension theorem. Weomit the details.age 5 J. M. Fraser The distance set problem , originating with the paper [F85], is a well-studied problem ingeometric measure theory. It was received a lot of attention in the literature in the lastfew years, see for example [F18, GIOW20, KS19, O17, S17, S19, S]. Given F ⊆ R n , the distance set of F is D ( F ) = {| x − y | : x, y ∈ F } ⊆ [0 , ∞ ) . The distance set problem is to understand the relationship between the dimensions of F and D ( F ). It is conjectured that if F ⊆ R n is Borel and dim H F > n/
2, then dim H D ( F ) = 1.This conjecture is open for all n >
2. The same conjecture can also be made with Hausdorffdimension replaced by Assouad dimension. This conjecture is also open, although it wasproved in [F18] that for F ⊆ R , dim A F > A D ( F ) = 1. We are ableto fully resolve the Assouad dimension version of the distance set problem in the plane,both dealing with the awkward ‘critical case’ dim A F = 1 and providing sharp estimatesfor sets with Assouad dimension less than 1. We emphasise that we do not required F tobe bounded or Borel. Theorem 3.1.
For all non-empty sets F ⊆ R , dim A D ( F ) > min { dim A F, } . Theorem 3.1 follows immediately from the more general Theorem 3.3 below. Theorem3.1 is sharp, as the following corollary shows. For comparison, it was already observed in[F18] that, for all s ∈ [0 , { dim A D ( F ) : F ⊆ R and dim A F s } = 1 . Corollary 3.2.
For all s ∈ [0 , , inf { dim A D ( F ) : F ⊆ R and dim A F > s } = s. Proof.
The lower bound ( > s ) follows from Theorem 3.1. The upper bound ( s ) followsby a standard construction: see, for example, [F18, Section 3.3.1]. Briefly, for s ∈ (0 , F ⊆ [0 ,
1] be a self-similar set generated by ⌈ N s ⌉ equally spaced homotheties withcontraction ratio 1 /N . This ensures that dim A F > s . Moreover, for V = span(1 , − ∈ G (2 , D ( F ) has Assouad dimension no more than that of π V ( F × F ),which is itself a self-similar set generated by 2 ⌈ N s ⌉ − /N . As N → ∞ , dim A D ( F ) approaches s .The next theorem considers the distance problem in R n for arbitrary n >
2. It showsthat the set of exceptions to (1.2) plays a role. Consider projections of sets of Assouaddimension s from R n onto m -dimensional subspaces and let θ ( s, n, m ) be the largest possibleHausdorff dimension of set of exceptions to (1.2), recall (2.1). Theorem 3.3.
For all non-empty sets F ⊆ R n , dim A D ( F ) > min { dim A F − θ, } where θ = θ (dim A F, n, . age 6 J. M. FraserThe proof of Theorem 3.3 requires some technical machinery we have not yet introduced.Therefore we delay the proof to Section 5. Theorem 3.1 follows from Theorem 3.3 togetherwith Orponen’s projection theorem [O, Theorem 1.1] which states that θ ( s, ,
1) = 0 for all s ∈ [0 , θ ( s, n, min { n − s, n + s − } . Combining this with Theorem 3.3 we get the following, which does not improve over knownresults, e.g. [F18, Theorem 2.5], but provides a somewhat different proof.
Corollary 3.4. If F ⊆ R n with dim A F > ( n + 1) / , then dim A D ( F ) = 1 . The bound (2.2) for θ ( s, n, m ) was proved by applying the bounds for the exceptionalset in the Marstrand-Mattila projection theorem (1.1). Orponen’s projection theorem isreason to believe that much better bounds are available in the Assouad dimension case.Indeed, if we could prove that θ ( n/ , n, n/ −
1, then all F ⊆ R n with dim A F > n/ A D ( F ) = 1. However, this is not true, at least for n = 3. Proposition 3.5.
For all n > , θ ( s, n, > n − for all s ∈ [1 , .Proof. Let V ∈ G ( n, n −
1) and E ⊆ V be contained in a line segment with dim A E = s − F = E × [0 , ⊆ R n . Clearly dim A F = s and for all V ∈ G ( n,
1) with V ⊆ V theprojection π V ( F ) is the image of E under a similarity (possible with contraction ratio 0).Therefore, for all such V , dim A π V ( F ) = s − < { s, } . The Hausdorff dimension of the set of such V is the same as that of G ( n − ,
1) which is n − A related problem is to consider pinned distance sets. Given x ∈ R n , the pinned distanceset of F ⊆ R n at x is D x ( F ) = {| x − y | : y ∈ F } . If x ∈ F , then D x ( F ) ⊆ D ( F ). Here the conjecture is that if F is Borel and dim H F > d/ x ∈ F such that dim H D x ( F ) = 1 (or even many pins). We arealso able to prove some results on pinned distance sets in the Assouad dimension setting. Theorem 3.6.
Let F ⊆ R n be a non-empty bounded set. For Lebesgue almost all x ∈ R n , dim A D x ( F ) > min { dim A F, } . Moreover, the set of exceptional x where this does not hold has Hausdorff dimension atmost θ (dim A E, n, n − | dim A F − | . age 7 J. M. Fraser Proof.
For t ∈ R n , consider the maps Π t : R n → R defined byΠ t ( x ) = | x − t | . Then, T z ( t ) = ker( J z Π t ) ⊥ = span( z − t ) ∈ G ( n, t = z . Since the preimage of V ∈ G ( n,
1) under T z is a line (with Hausdorffdimension 1), the triple ( { Π t : t ∈ R n } , H u +1 , H u ) is a generalised family of projections of R n of rank 1 for all u >
0. The results follow by applying Corollary 2.3 (with u = n − u > θ (dim A F, n, t ( F ) = D t ( F ).The quantitative bound comes from (2.2).We can upgrade this result in the planar case since θ ( s, ,
1) = 0 for all s ∈ [0 , Corollary 3.7.
Let F ⊆ R be a non-empty bounded set. For all x ∈ R outside of a setof exceptions of Hausdorff dimension at most , dim A D x ( F ) > min { dim A F, } . Therefore, if dim H F > , then there exists x ∈ F such that dim A D x ( F ) = 1 . Shmerkin [S19] proved that if F ⊆ R is a Borel set with equal Hausdorff and packingdimension strictly larger than 1, then there exists x ∈ F such that dim H D x ( F ) = 1. It is also natural to consider the distance set (and pinned distance set) problem with respectto norms other than the Euclidean norm. That is, given a norm k · k on R n , the distanceset of F ⊆ R n with respect to k · k is D k·k ( F ) = {k x − y k : x, y ∈ F } with the obvious analogous definition of pinned distance sets D k·k x ( F ). Whether or not weexpect the same results to hold turns out to depend on the curvature of the unit ball inthe given norm. Theorems 3.1, 3.3, 3.6 and Corollary 3.7 hold in this more general settingprovided the boundary of the unit ball ∂B is a C manifold and the associated Gauss mapcannot decrease Hausdorff dimension (that is, dim H g ( E ) > dim H E for all E ⊆ ∂B , where g : ∂B → S n − is the Gauss map). For example, this holds if ∂B is a C manifold withnon-vanishing Gaussian curvature, since in that case the Gauss map is a diffeomorphism,see [G02, Corollary 3.1].Let Π k·k t : R n → R denote the pinned distance map with respect to a general norm,that is, Π k·k t ( x ) = k x − t k , and let T z ( t ) = ker( J z Π k·k t ) ⊥ . If the boundary of the unit ball ∂B is C , then the restriction of T z to ( ∂B + z ) coincides with the Gauss map (identifyingantipodal points in S n − and then identifying with G ( n, H u +1 ◦ T − z ≪ H u age 8 J. M. Fraserfor all u >
0. This observation allows the proof of Theorem 3.6 (and Corollary 3.7) togo through in this more general setting. The proof of Theorem 3.3 (and Theorem 3.1) isdeferred until Section 5 and so we also defer discussion of its extension to general norms.The assumption of non-vanishing Gaussian curvature is natural when studying distancesets. Indeed, for certain “flat norms” the analogous results do not hold, see [F04]. See recentexamples [GIOW20, S] where results are obtained for the Hausdorff dimension of distancesets under the assumption that the unit ball is C ∞ and C , respectively, in addition tohaving non-vanishing Gaussian curvature. It is perhaps noteworthy that we only require C regularity and a weaker condition on the Gauss map. For example, our techniques allowfor the Gaussian curvature to vanish on a countable set of points. Radial projections are perhaps the most natural family of projections alongside orthogonalprojections. Given t ∈ R n , the radial projection π t maps R n \ { t } onto the boundary of thesphere centred at t with radius 1. Specifically, π t ( x ) ∈ t + S n − is defined by π t ( x ) = x − t | x − t | + t and we define π t ( t ) = t for convenience. Radial analogues of results such as the Marstrand-Mattila projection theorem are known and turn out to be important in their own right ina variety of settings. For example, Orponen’s radial projection theorem [O19] has proveda useful tool in in studying the distance set problem, see [GIOW20, KS19]. Recall thedefinition of θ from (2.1). Theorem 3.8.
Let F ⊆ R n be a non-empty bounded set. For Lebesgue almost all t ∈ R n , dim A π t ( F ) > min { dim A F, n − } . Moreover, the set of exceptional t ∈ R n where this does not hold has Hausdorff dimensionat most θ (dim A F, n, n − min { dim A F + 1 , n − − dim A F } .Proof. For all z ∈ R n and t = z , π t is smooth on B ( z, | z − t | /
2) and T z ( t ) = ker( J z π t ) ⊥ = span( z − t ) ⊥ ∈ G ( n, n − . Since the preimage of V ∈ G ( n, n −
1) under T z is again a line, the triple ( { π t : t ∈ R n } , H u +1 , H u ) is a generalised family of projections of R n of rank n −
1. The results followby applying Corollary 2.3 (with u = n −
1) and Corollary 2.4 (with u > θ (dim A F, n, n − S n − can be replaced by any smooth enough ( n − S ⊆ R n be a simply connected compact ( n − C manifold, with the property that for all x ∈ R n \ { } the intersection { λx : λ > } ∩ S age 9 J. M. Fraseris a singleton, which we denote by S ( x ). Then the family of radial projections onto S withcentre t ∈ R n given by π S t ( x ) = S ( x − t ) + t also satisfies the conclusion of Theorem 3.8. Moreover, the exceptional set does not dependon S and so the conclusion holds for all S simultaneously.We obtain a sharp result concerning the dimension of the exceptional set in Theorem3.8 in the planar case since θ ( s, ,
1) = 0 for all s ∈ [0 ,
2] by Orponen’s projection theorem[O].
Corollary 3.9.
Let F ⊆ R be a non-empty bounded set. Then dim A π x ( F ) > min { dim A F, } for all x ∈ R outside of a set of exceptions of Hausdorff dimension at most 1. Corollary 3.9 is clearly sharp since a line segment will radially project to a single pointfor all t in the affine span of the line segment. ‘Sum-product’ results in additive combinatorics refer to a wide range of phenomena regard-ing the ‘independence’ of multiplication and addition. For example, for a set F ⊂ (0 , F F = { xy : x, y ∈ F } and the sumset F + F = { x + y : x, y ∈ F } to be small simultaneously. If F is finite, then size means cardinality and this statementis made precise by the Erd˝os-Szemer´edi theorem. If F is infinite then it is natural todescribe size in terms of dimension. The following is a sum-product type result for Assouaddimension, where we are also able to consider independence of other operations such asaddition and exponentiation. For F ⊆ (0 , ∞ ), we write F F = { x y : x, y ∈ F } . Theorem 3.10.
Let F ⊆ R be a non-empty bounded set with dim H F > . Then dim A ( F F + F ) > min { A F, } , and, if F ⊆ (0 , ∞ ) , dim A ( F F + F ) > min { A F, } . Proof.
For t ∈ R , consider the family of projections Π t : R → R defined byΠ t ( x, y ) = tx + y. age 10 J. M. FraserApplying Corollary 2.5 with s = u to the cartesian product F × F = { ( x, y ) : x, y ∈ F } (not to be confused with F F ) we getdim H { t : dim A Π t ( F × F ) < min { dim A ( F × F ) , }} = 0 . Since dim H F >
0, there must exist t ∈ F such thatdim A Π t ( F × F ) > min { dim A ( F × F ) , } = min { A F, } . The result follows since Π t ( F × F ) = tF + F ⊆ F F + F . The fact that dim A ( F × F ) =2 dim A F can be found in, for example, [L98, Theorem A.5 (5)]. The second result is provedsimilarly, but the details are more involved. For t >
0, consider the family of projectionsΠ t : R → R defined by Π t ( x, y ) = t x + y. Here T ( x,y ) ( t ) = ker( J ( x,y ) Π t ) ⊥ = span (cid:16) , t − x log( t ) (cid:17) ∈ G (2 , t > , −∞ ) = span(0 ,
1) when t = 1). Although T ( x,y ) : (0 , ∞ ) → G (2 ,
1) is not generally surjective or injective, we stillhave H s ◦ T − x,y ) ≪ H s for all s >
0. Therefore, by applying Corollary 2.5 to F × F ,dim H { t > A Π t ( F × F ) < min { dim A ( F × F ) , }} = 0 . Since dim H F > F ⊆ [0 , ∞ ), there must exist t ∈ F such thatdim A Π t ( F × F ) > min { dim A ( F × F ) , } = min { A F, } . The result follows since Π t ( F × F ) = t F + F ⊆ F F + F .This example was partly motivated by Orponen’s paper [O17]. Orponen [O17, Corollary1.5] proved that if F ⊆ R is compact, Ahlfors-David regular, and has dim H F > /
2, thendim P ( F F + F F − F F − F F ) = 1 , where dim P denotes packing dimension. We are able to provide a much stronger result, butwith packing dimension replaced by Assouad dimension. Notably, the set F need not beAhlfors-David regular, we consider the much smaller set F F + F , and we obtain estimatesfor sets with arbitrarily small dimension. We note that since the family of projections usedto handle F F + F in Theorem 3.10 are orthogonal, this result could be deduced directly fromOrponen’s projection theorem. The set F F + F requires our nonlinear theorem, however.Finally, we observe that many other sets constructed from F can be handled in this way— or even sets constructed from a collection of sets, rather than the single set F . We leavethe details to the interested reader.age 11 J. M. Fraser As a final application we revisit one of the situations where Peres and Schlag [PS00] wereable to apply their nonlinear projection theorem. Given two non-empty sets
E, F ⊆ R with sufficient ‘arithmetic independence’, one might hope for dim( E + F ) = min { dim E +dim F, } . This can fail for many reasons but if we parameterise F in a transversal enoughway, then we can recover this formula generically. Following [PS00], for λ ∈ (0 , /
2) we let F λ = (cid:26) X n > i n λ n : i n ∈ { , } (cid:27) and consider E + F λ for generic λ . For all λ ∈ (0 , / F λ is a compact self-similar Cantorset with dim H F λ = dim A F λ = − log 2 / log λ . The following result also holds for moregeneral homogeneous Cantor sets, but we omit the details. Theorem 3.11.
Let E ⊆ R be non-empty. Then, for almost all λ ∈ (0 , / , dim A ( E + F λ ) > min { dim A E + dim A F λ , } . Moreover, the set of exceptional λ in a given interval ( a, b ) ⊆ (0 , / for which this doesnot hold has Hausdorff dimension at most dim A E + dim A F b . One of the distinguishing features of this result is that the generic dimension bounddepends on the parameter λ . The proof will be a straightforward combination of ourapproach and the result of Peres and Schlag. Nevertheless, we delay the proof until Section6. The tangent structure of a set is intimately related to the Assouad dimension and it is viathe tangent structure that we will prove Theorem 2.2. Mackay and Tyson [MT10] pioneeredthe theory of weak tangents in the context of Assouad dimension. Weak tangents are limitsof sequences of blow-ups of a given set with respect to the Hausdorff metric. Rather thanuse weak tangents directly, it is more convenient for us to use the non-symmetric Hausdorffdistance defined by ρ H ( A, B ) = sup a ∈ A inf b ∈ B | a − b | for non-empty closed sets A, B ⊆ R n . The Hausdorff metric is then defined as d H ( A, B ) = max { ρ H ( A, B ) , ρ H ( B, A ) } for non-empty compact sets A, B ⊆ R n . In what follows we choose to approximate using ρ H rather than d H . An alternative would have been to approximate using d H via subsets ,but we found this more cumbersome. This approach was used, for example, in [FHOR15,Definition 3.6] with the terminology weak pseudo tangent . Another minor variation we makeon the usual theory of weak tangents is to allow some flexibility in the blow-ups: they needage 12 J. M. Frasernot be via strict similarities. This approach was used, for example, in [F14b, Proposition7.7] with the terminology very weak tangents . To simplify exposition and terminology, wesimply refer to tangents . We write B ( x, r ) for the closed ball centred at x ∈ R n with radius r > Definition 4.1.
Let
E, F ⊆ R n be closed sets with E ⊆ B (0 , . Suppose there exists asequence of maps S k : R n → R n and constants a k , b k > with sup k ( b k /a k ) < ∞ such that a k | x − y | | S k ( x ) − S k ( y ) | b k | x − y | for all x, y ∈ S − k ( B (0 , and suppose that ρ H ( E, S k ( F )) → as k → ∞ . Then we call E a tangent to F . If each S k is a homothety, that is, S k ( x ) = c k x + t k for some c k > and t k ∈ R n , and c k → ∞ , then we call E a simple tangent to F . The maps S k in Definition 4.1 blow-up the set F around z k = S − k (0). If the limit z = lim k →∞ z k ∈ R n exists, then we call z the focal point of E . Note that if F is compactand E is a simple tangent to F , then we may assume (by taking a subsequence if necessary)that the focal point exists and, moreover, is a point in F . The following is a minor varianton a result of Mackay and Tyson [MT10, Proposition 6.1.5]. Theorem 4.2.
Let F ⊆ R d be closed and E ⊆ R d be a tangent to F . Then dim A F > dim A E . The following result of K¨aenm¨aki, Ojala and Rossi [KOR18, Proposition 5.7] shows thatTheorem 4.2 has a useful converse.
Theorem 4.3.
Let F ⊆ R d be closed and non-empty. Then there exists a compact set E ⊆ R d with dim H E = dim A F such that E is a simple tangent to F . The key technical result required to prove Theorem 2.2 is the following proposition. It statesthat there is an appropriately chosen orthogonal projection of a simple tangent, which is atangent to a given nonlinear projection.
Proposition 4.4.
Let F ⊆ R n be non-empty and compact. Suppose E is a simple tangentto F with focal point z ∈ F . Further suppose that t ∈ Ω is such that Π t is C and ofconstant rank m > in a neighbourhood of z . Then π V ( E ) is a tangent to Π t ( F ) for V = ker( J z Π t ) ⊥ ∈ G ( n, m ) . Before proving Proposition 4.4, we provide some preliminary results. We may assumefor convenience that E ⊆ B (0 , / S k be a sequence of homothetic similarities of R n such that ρ H ( E, S k ( F )) → . age 13 J. M. FraserWrite c k > S k and t k ∈ R n for the associated translation. Let z k ∈ F be such that S k ( B ( z k , c − k )) = B (0 ,
1) and z = lim k →∞ z k ∈ F be the focal point of E . (Note that 0 = S k ( z k ) = c k z k + t k .) Let V = ker( J z Π t ) ⊥ and V k = ker( J z k Π t ) ⊥ , notingthat V k , V ∈ G ( n, m ) for large enough k by the differentiability assumption. Moreover, V k → V in the Grassmannian metric d G , defined by d G ( U, U ′ ) = d H (cid:16) U ∩ B (0 , , U ′ ∩ B (0 , (cid:17) for U, U ′ ∈ G ( n, m ). This convergence is guaranteed by the assumption that Π t is continu-ously differentiable in a neighbourhood of z , and therefore ker( J z ′ Π t ) ⊥ varies continuouslyfor z ′ sufficiently close to z .There exists a constant c = c ( z, t ) > k sufficiently large and all x, y ∈ ker( J z k Π t ) ⊥ , | ( J z k Π t )( x ) − ( J z k Π t )( y ) | > c k J z k Π t k| x − y | , (4.1)where k · k denotes the operator norm. This can be guaranteed since z k → z , Π t is continu-ously differentiable in a neighbourhood of z , and J z Π t is injective and linear on ker( J z Π t ) ⊥ .Fix ε ∈ (0 , /
10) satisfying 0 < ε < ( c/ k J z Π t k (4.2)where c = c ( z, t ) > z and t are fixed.Define U k : Π t ( B ( z k , c − k )) → R n by U k = S k ◦ U k where U k : Π t ( B ( z k , c − k )) → B ( z k , c − k ) is defined by letting U k ( u ) be the unique point in the intersectionΠ − t ( u ) ∩ ( V k + z k ) ∩ B ( z k , c − k ) . Lemma 4.5.
The map U k is well-defined for sufficiently large k .Proof. Throughout this proof we restrict Π t to a neighbourhood of z such that it is C and of constant rank m . By the implicit function theorem, the level set Π − t ( u ) isa simply connected ( n − m )-dimensional C manifold which intersects B ( z k , c − k ) since u ∈ Π t ( B ( z k , c − k )). This follows by expressing the action of Π t near z in local coordinates.Moreover, since Π t is differentiable, vectors v in the tangent space T x Π − t ( u ) at x ∈ Π − t ( u )coincide with directional derivatives of Π t at x in direction v . For the manifold Π − t ( u ) tointersect ( V k + z k ) more than once, or not at all, inside B ( z k , c − k ) we would require thetangent spaces of Π − t ( u ) at points inside B ( z k , c − k ) to differ from ker( J z k Π t ) by morethan 1/100 (in the Grassmannian metric, say). This is impossible for large enough k sinceΠ t is continuously differentiable in a neighbourhood of z .We will use the maps U k to show that π V ( E ) ⊆ B (0 , ∩ V is a tangent to Π t ( F ).Therefore we must show these maps satisfy the conditions from Definition 4.1. Since S k is a homothety, it is sufficient to demonstrate that U k satisfies the conditions. This is thecontent of the next lemma. Note that we only need to consider points which map into B (0 ,
1) under U k , which is consistent with the domain of U k being Π t ( B ( z k , c − k )). We mayextend U k (and thus U k ) to a mapping on the whole of R n if we wish, but this is not reallynecessary.age 14 J. M. Fraser Lemma 4.6.
For sufficiently large k , for all x, y ∈ Π t S − k ( B (0 , t ( B ( z k , c − k ))1(2 + c ) k J z Π t k | x − y | | U k ( x ) − U k ( y ) | c k J z Π t k | x − y | where c is the constant from (4.1) .Proof. Since Π t is continuously differentiable in a neighbourhood of z , we may assume k islarge enough to ensure | Π t ( b ) − Π t ( a ) − ( J a Π t )( b − a ) | ε | b − a | (4.3)for all a, b ∈ B ( z k , c − k ). We may also assume k is large enough to ensure(1 / k J z Π t k k J a Π t k k J z Π t k (4.4)for all a ∈ B ( z k , c − k ). This estimate can be achieved because J z Π t is continuous at z and k J z Π t k >
0. These facts are guaranteed since Π t is continuously differentiable ina neighbourhood of z and J z Π t has strictly positive rank, respectively. Finally, we mayassume k is large enough to guarantee1 / | ( J z k Π t )( x − y ) || ( J x Π t )( x − y ) | x, y ∈ B ( z k , c − k ) ∩ ( V k + z k ). This can be achieved since J z k Π t is linear and injectiveon V k and J x Π t continuous in x in a neighbourhood of z . In particular, J z k Π t → J z Π t .Fix distinct x, y ∈ Π t ( B ( z k , c − k )). Since U k ( x ) − U k ( y ) ∈ V k , by (4.1) and (4.4), | ( J z k Π t )( U k ( x ) − U k ( y )) | > c k J z k Π t k| U k ( x ) − U k ( y ) | > ( c/ k J z Π t k| U k ( x ) − U k ( y ) | . (4.6)Moreover, using the fact that U k is injective, | ( J z k Π t )( U k ( x ) − U k ( y )) | = | ( J z k Π t )( U k ( x ) − U k ( y )) || ( J U k ( x ) Π t )( U k ( x ) − U k ( y )) | | ( J U k ( x ) Π t )( U k ( x ) − U k ( y )) | | Π t U k ( x ) − Π t U k ( y ) | + 2 ε | U k ( x ) − U k ( y ) | by (4.3) and (4.5)= 2 | x − y | + 2 ε | U k ( x ) − U k ( y ) | (4.7)since Π t U k is the identity on Π t ( B ( z k , c − k )). Combining (4.6) and (4.7) and using (4.2)yields | U k ( x ) − U k ( y ) | c/ k J z Π t k − ε | x − y | c k J z Π t k | x − y | age 15 J. M. Fraseras required. The lower bound is similar. By the definition of the operator norm k · k and(4.4), | ( J z k Π t )( U k ( x ) − U k ( y )) | k J z k Π t k| U k ( x ) − U k ( y ) | k J z Π t k| U k ( x ) − U k ( y ) | . (4.8)Moreover, using the fact that U k is injective, | ( J z k Π t )( U k ( x ) − U k ( y )) | = | ( J z k Π t )( U k ( x ) − U k ( y )) || ( J U k ( x ) Π t )( U k ( x ) − U k ( y )) | | ( J U k ( x ) Π t )( U k ( x ) − U k ( y )) | > (1 / | Π t U k ( x ) − Π t U k ( y ) | − ( ε/ | U k ( x ) − U k ( y ) | by (4.3) and (4.5)= (1 / | x − y | − ( ε/ | U k ( x ) − U k ( y ) | (4.9)since Π t U k is the identity on Π t ( B ( z k , c − k )). Combining (4.8) and (4.9) yields | U k ( x ) − U k ( y ) | > / k J z Π t k + ε/ | x − y | > c ) k J z Π t k | x − y | as required.The next result is a technical approximation which says that close to z k the composition U k Π t behaves very much like orthogonal projection onto V + z k . Lemma 4.7.
For sufficiently large k > , sup w ∈ B ( z k ,c − k ) | S − k π V S k ( w ) − U k Π t ( w ) | c − k ε. Proof.
Let w ∈ B ( z k , c − k ) and write u = Π t ( w ). Then U k Π t ( w ) = Π − t ( u ) ∩ ( V k + z k ) ∩ B ( z k , c − k ). For sufficiently large k , the tangent spaces of the manifold Π − t ( u ) are in an ε -neighbourhood of ker( J z k Π t ) = V ⊥ k (in the Grassmannian metric) and since | w − z k | c − k we conclude that | U k Π t ( w ) − π V k ( w − z k ) − z k | εc − k for large enough k . Moreover, since V k → V in d G , for sufficiently large k we have | π V ( w − z k ) + z k − π V k ( w − z k ) − z k ) | d G ( V k , V ) | w − z k | εc − k . Finally, π V ( w − z k ) + z k = S − k π V S k ( w ) and the result follows.Next we provide a pair of simple algebraic identities.age 16 J. M. Fraser Lemma 4.8.
For all integers k and all w ∈ R n S k π V S − k ( w ) = π V ( w ) − π V ( t k ) + t k (4.10) and S k π V ( w ) + π V ( t k ) − t k = c k S − k π V S k ( w ) . (4.11) Proof.
These identities follow immediately by applying the definition of S k and using thefact that linear homotheties and orthogonal projections commute.We are now ready to prove Proposition 4.4 Proof.
Fix x ∈ π V ( E ). Choose k large enough to guarantee that the conclusion of Lemma4.7 holds and also that ρ H ( E, S k ( F )) ε/ . (4.12)Choose y ∈ S k ( F ) ∩ B (0 ,
1) such that | x − π V ( y ) | ε (4.13)which we may do by first applying (4.12) and then the fact that orthogonal projections donot increase distances. Then | x − U k Π t S − k ( y ) | = | x − S k U k Π t S − k ( y ) | | x − S k π V S − k ( y ) − π V ( t k ) + t k | + | S k π V S − k ( y ) + π V ( t k ) − t k − S k U k Π t S − k ( y ) | = | x − π V ( y ) | by (4.10)+ | S k π V S − k ( y ) + π V ( t k ) − t k − c k U k Π t S − k ( y ) | = | x − π V ( y ) | + c k | S − k π V S k S − k ( y ) − U k Π t S − k ( y ) | by (4.11) ε + c k (2 c − k ε )by (4.13) and Lemma 4.7. Since S − k ( y ) ∈ F ∩ B ( z k , c − k ) ⊆ F , we have proved that, for allsufficiently large k , ρ H ( π V ( E ) , U k Π t F ) ε. Since, by Lemma 4.6, U k satisfies the conditions required in Definition 4.1 for sufficientlylarge k , it follows that π V ( E ) is a tangent to Π t ( F ), completing the proof.age 17 J. M. Fraser Theorem 2.2 follows succinctly from Proposition 4.4. First suppose F is closed. ApplyTheorem 4.3 to obtain a simple tangent E with focal point z ∈ F satisfying dim H E =dim A F . Proposition 4.4, the differentiability assumption in Definition 2.1, and Theorem4.2 imply that for µ almost all t ∈ Ωdim A Π t ( F ) > dim A π V ( t ) ( E )for V ( t ) = T z ( t ) = ker( J z Π t ) ⊥ ∈ G ( n, m ). Sincedim A π V ( E ) > essinf V ∼ P dim A π V ( E )for P almost all V ∈ G ( n, m ) and µ ◦ T − z ≪ P (the absolute continuity assumption inDefinition 2.1), it follows thatdim A π V ( t ) ( E ) > essinf V ∼ P dim A π V ( E )holds for µ almost all t ∈ Ω. Therefore, since dim H E = dim A F ,dim A Π t ( F ) > inf E ⊆ R n :dim H E =dim A F essinf V ∼ P dim A π V ( E )holds for µ almost all t ∈ Ω, proving the theorem for closed F . However, if F is not closed,then Π t ( F ) ⊇ Π t ( F ) since Π t is continuous. Therefore, since Assouad dimension is stableunder taking closure, dim A Π t ( F ) = dim A Π t ( F ) > dim A Π t ( F )and the desired result follows by applying the result for closed sets. A key step in the proof of Theorem 3.3 will be to relate pinned distance sets and radialprojections via radial product sets. Given X ⊆ S n − and Y ⊆ R , we define the radialproduct of X and Y to be the set X ⊗ Y = { xy : x ∈ X, y ∈ Y } ⊆ R n . The following is more general than we need. We write dim B for the upper box dimensionand note that for bounded sets E ⊆ R n dim H E dim B E dim A E. For concreteness, the upper box dimension of a bounded set E is the infimum of α > C > r > E may be covered by fewer than Cr − α sets of diameter r .age 18 J. M. Fraser Lemma 5.1.
For X ⊆ S n − and bounded Y ⊆ R , dim H ( X ⊗ Y ) dim H X + dim B Y. Proof.
This is straightforward but we include the details due to its importance. Fix s > dim H X and t > dim B Y . Let ε > δ > { U i } i be a finite or countable δ -cover of X such that X i | U i | s ε. Consider the ‘wedge’ W i = { xy : x ∈ X ∩ U i , y ∈ Y } . By the definition of upper boxdimension, there exists a uniform constant C > W i may be covered by fewerthan C | U i | − t sets of diameter | U i | . Taking the union of these covers over all i yields a δ -cover { V j } j of X ⊗ Y satisfying X j | V j | s + t X i | U i | s + t C | U i | − t Cε which proves that dim H ( X ⊗ Y ) s + t , and thus the lemma.It is immediate that for all sets E ⊆ R n and z ∈ R n E ⊆ ( π z ( E ) − z ) ⊗ D z ( E ) + z. (5.1)Indeed, for x ∈ E ( π z ( x ) − z ) ⊗ D z ( x ) + z = x. Therefore Lemma 5.1 yieldsdim H E dim H π z ( E ) + dim B D z ( E ) . (5.2)We are now ready to prove Theorem 3.3. Proof.
It was proved in [F18, Lemma 3.1] that if F ⊆ R n is a closed set and E a simpletangent to F , then dim A D ( F ) > dim A D ( E ) . Therefore it is sufficient to work with tangents of F . Assume for now that F is closed andapply Theorem 4.3 to obtain a compact simple tangent E to F withdim H E = dim A F. Apply Theorem 4.3 a second time to obtain a compact simple tangent E ′ to E withdim H E ′ = dim A E = dim A F and let z ∈ E be the focal point of E ′ . Let E ⊆ G ( n,
1) be the set of exceptions to (1.2)applied to E ′ . By definition E has Hausdorff dimension at most θ = θ (dim A F, n, H π z ( E ) > θ . Since dim H E θ , there must exist x ∈ E suchthat span( z − x ) / ∈ E . Proposition 4.4 implies that π span( z − x ) ( E ′ ) is a tangent to D z ( E ).Therefore, applying Theorem 4.2,dim A D ( F ) > dim A D ( E ) > dim A D z ( E ) > dim A π span( z − x ) ( E ′ ) > min { dim A E ′ , } = min { dim A F, } . Case 2: Suppose dim H π z ( E ) θ . It follows from (5.2) thatdim H E dim H π z ( E ) + dim B D z ( E ) θ + dim A D z ( E )and thereforedim A D ( F ) > dim A D ( E ) > dim A D z ( E ) > dim H E − θ = dim A F − θ. Therefore we have proved the desired result for closed sets F . If F is not closed, then D ( F ) ⊇ D ( F ) since the map from R n × R n to R defined by ( x, y )
7→ | x − y | is continuous.Therefore, since Assouad dimension is stable under taking closure,dim A D ( F ) = dim A D ( F ) > dim A D ( F )and the desired result follows by applying the result for closed sets. The proof given in Section 5 goes through almost verbatim if the distance set is definedvia a general norm k · k with the property that the boundary of the unit ball ∂B is a C manifold and the associated Gauss map cannot decrease Hausdorff dimension, see Section3.1.2. In the definition of radial product, S n − is replaced by ∂B and then (5.1) holds withthe radial projection and pinned distance maps taken with respect to the norm k · k . Theproof of [F18, Lemma 3.1] goes through almost unchanged in the setting of general normsand therefore we can reduce to tangents E and E ′ in exactly the same way. Finally, writingΠ k·k t : R n → R for the pinned distance map with respect to k · k , the case 1 assumptiondim H π z ( E ) > θ still guarantees existence of x ∈ E such that T z ( x ) = ker( J z Π k·k x ) / ∈ E . Thisis because the restriction of T z to ( ∂B + z ) coincides with the Gauss map g : ( ∂B + z ) → S n − (upon identification of antipodal points in S n − and then identification with G ( n, Apply Theorem 4.3 to obtain a simple tangent E ′ to E with dim H E ′ = dim A E and let z ∈ E be the focal point of E ′ . It is straightforward to see that F λ is itself a simple tangentto F λ with focal point 0. Therefore E ′ × F λ is a simple tangent to E × F λ with focal point( z,
0) for all λ ∈ (0 , / V = span(1 , ∈ G (2 , π V ( E ′ × F λ ) is a tangent to π V ( E × F λ ) and therefore, by Theorem 4.2,dim A ( E + F λ ) = dim A π V ( E × F λ ) > dim H π V ( E ′ × F λ ) = dim H ( E ′ + F λ )age 20 J. M. Fraserfor all λ ∈ (0 , / H ( E ′ + F λ ) = min { dim H E ′ + dim H F λ , } = min { dim A E + dim A F λ , } for almost all λ ∈ (0 , /
2) and even all λ ∈ ( a, b ) ⊆ (0 , /
2) outside of a set of exceptionsof Hausdorff dimension at most dim H E ′ + dim H F b = dim A E + dim A F b , completing theproof. Acknowledgements
The author was supported by an
EPSRC Standard Grant (EP/R015104/1) and a
Lever-hulme Trust Research Project Grant (RPG-2019-034).
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