A dichotomy for bounded displacement equivalence of Delone sets
aa r X i v : . [ m a t h . M G ] O c t A DICHOTOMY FOR BOUNDED DISPLACEMENT ANDCHABAUTY-FELL CONVERGENCE OF DISCRETE SETS
YOTAM SMILANSKY AND YAAR SOLOMON
Abstract.
We prove that in every compact space of Delone sets in R d which is minimalwith respect to the action by translations, either all Delone sets are uniformly spread, orcontinuously many distinct bounded displacement equivalence classes are represented,none of which contains a lattice. The implied limits are taken with respect to theChabauty-Fell topology on the space of closed subsets of R d . This topology coincideswith the standard local topology in the finite local complexity setting, and it follows thatthe dichotomy holds for all minimal spaces of Delone sets associated with well-studiedconstructions such as cut-and-project sets and substitution tilings, whether or not finitelocal complexity is assumed.A main step in the proof is a result concerning Delone sets as limits of convergingsequences of finite patches with respect to the Chabauty-Fell topology, under the assump-tion of minimality. In the infinite local complexity setting, information on a convergingsequence does not immediately imply information regarding finite patches in the limitDelone set, and we provide sufficient conditions under which certain qualitative andquantitative information can be deduced. Introduction
A set Λ Ă R d is called a Delone set if it is both uniformly discrete and relatively dense ,that is, if there are constants r, R ą r contains at most onepoint of Λ and Λ intersects every ball of radius R . We refer to r and R as the separationconstant and the packing radius of Λ, respectively. Two Delone sets Λ , Γ Ă R d are said tobe bounded displacement (BD) equivalent if there exists a bijection φ : Λ Ñ Γ satisfyingsup x P Λ } x ´ φ p x q} ă 8 . Such a mapping φ is called a BD-map . All lattices in R d with the same covolume areBD-equivalent, and a Delone set Λ is called uniformly spread if it is equivalent to a lattice,or equivalently, if there is a BD-map φ : Λ Ñ α Z d , for some α ą ρ on R d and consider the space Cl p R d q of closed subsets of R d . The Chabauty-Fell topology on Cl p R d q is the topology induced by the metric D p Λ , Λ q def “ inf ˜ ε ą ˇˇˇ Λ X B p , { ε q Ă Λ p` ε q Λ X B p , { ε q Ă Λ p` ε q + Y t u ¸ , (1.1)where B p x , R q is the open ball of radius R ą x P R d with respect to themetric ρ , and A p` ε q is the ε neighborhood of the set A . This metric was introduced byChabauty [Ch] for Cl p R d q , as well as for a more general setting, and later extended by Fell[Fe], see also [LSt] for the relation to the Hausdorff metric. In this work we only considermetrics ρ that are determined by norms on R d , and although different choices of normsresult in different metrics D , they all define the same Chabauty-Fell topology. We remark that in the aperiodic order literature, this topology is often referred to as the local rubbertopology , see e.g. [BG, § R d are elements of Cl p R d q , and we may consider compact spaces of Delonesets, where the implied limits are taken with respect to the Chabauty-Fell topology. Sucha space X of Delone sets in R d is minimal with respect to the R d action by translationsif the orbit closure of every Delone set Λ P X is dense in X .Denote the cardinality of the set of BD-equivalence classes represented in X by BD p X q .The following dichotomy is our main result. Theorem 1.1.
Let X be a space of Delone sets in R d , and assume it is compact withrespect to the Chabauty-Fell topology and minimal with respect to the action of R d bytranslations. Then either (1) there exists a uniformly spread Delone set in X (and so every Λ P X is uniformlyspread and BD p X q “ ),or (2) BD p X q “ ℵ ,where ℵ denotes the cardinality of the continuum. Observe that the minimality assumption is essential, as shown by the following simpleexample. Consider Λ “ p´ N q \ t u \ N , a Delone set in R . Then the orbit closure X of Λ under translations by R and with respect to the Chabauty-Fell topology, consists oftranslations of Λ, the orbit closure of Z and the orbit closure of 2 Z . Therefore BD p X q “ X is not minimal.Let us remark that the implication in the brackets of (1) of Theorem 1.1 is a directconsequence of [La, Theorem 1.1], see also [FG, Theorem 3.2] for a sketch of a similarproof that holds for general minimal spaces of Delone sets. In addition, we note that auniformly discrete set in R d with separation constant r ą r Z d , hence the upper bound BD p X q ď ℵ is trivial.Recall that a Delone set Λ is of finite local complexity (FLC) if for every R ą R in Λ up to translationsis finite. For such sets the orbit closure under translations is sometimes called the hull ,and every Delone set in the hull is also FLC. The hull itself is then called FLC, and theChabauty-Fell topology on X coincides with the standard local topology , see [BG, § §
7] in the strongestpossible way.In addition to Theorem 1.1, we establish two results that may be of interest in theirown right. In § § D DICHOTOMY AND C-F CONVERGENCE 3 previously considered mainly in the context of uniformly spread point sets, see e.g. [DO1,DO2, La] and [DSS]. More recently, BD-equivalence emerged as an important objectof study for Delone sets that appear in the study of mathematical quasicrystals andaperiodic order, see [BG] for a comprehensive introduction to such constructions. Forcut-and-project sets, BD-equivalence was studied in [HKW], and links to the notions of bounded remainder sets and pattern equivariant cohomology appeared in [FG, HK, HKK]and in [KS1, KS2], respectively. For Delone sets associated with substitution tilings,sufficient conditions for a set to be uniformly spread were provided in [ACG], [S1] and[S2]. In addition, for the multiscale substitution tilings introduced by the authors in [SS],it was shown that any Delone set associated with an incommensurable tiling cannot beuniformly spread.Recently, questions regarding BD-non-equivalence between two Delone sets were con-sidered in [FSS], and a sufficient condition for BD-non-equivalence was established. Itwas later shown in [S3] that if the eigenvalues and eigenspaces of the substitution ma-trix satisfy a certain condition, then the tiling space contains continuously many distinctBD-classes.The following less restrictive equivalence relation on Delone sets is often studied inparallel to the BD-equivalence relation. We say that two Delone sets Λ and Γ are biLips-chitz equivalent if there exists a biLipschitz bijection between them. Namely, a bijection ϕ : Λ Ñ Γ and a constant C ě @ x , y P Λ 1 C ď } ϕ p x q ´ ϕ p y q}} x ´ y } ď C. It was shown by Burago and Kleiner [BK] and independently by McMullen [McM], thatthere exist Delone sets in R d , d ě
2, that are not biLipschitz equivalent to Z d . Magazinovshowed in [Mag] that there are continuously many Delone sets that are pairwise non-biLipschitz equivalent. It would be interesting to obtain an analogue of our Theorem 1.1in this context. Question.
Does Theorem 1.1 hold if one replaces the BD-equivalence relation by biLip-schitz equivalence?1.1.
Consequences of Theorem 1.1.
Theorem 1.1 directly implies that BD p X q “ ℵ for many special families of minimal spaces of Delone sets which are central in the theoryof aperiodic order, and for which the BD-equivalence relation was previously considered.1.1.1. Substitution tilings:
For primitive substitution tilings of R d , we denote by λ ą| λ | ě . . . ě | λ n | the eigenvalues of the substitution matrix, and we let t ě λ t contains non-zero vectors whose sum ofcoordinates is not zero. Under the assumption that tiles are bi-Lipschitz homeomorphicto closed balls, it was shown in [S2, Theorem 1.2 (I)] that if | λ t | ą λ p d ´ q{ d (1.2)then the Delone sets corresponding to the tilings in the tiling space are not uniformlyspread. Under the assumption (1.2) and an additional assumption regarding the existenceof certain patches, it was recently shown in [S3] that BD p X q “ ℵ . Given the aboveresult of [S2], and since substitution tiling spaces are minimal (see [BG]), the followingstrengthening of the main result of [S3] is a direct consequence of our Theorem 1.1. YOTAM SMILANSKY AND YAAR SOLOMON
Corollary 1.2.
Let X be a primitive substitution tiling space with tilings by tiles thatare bi-Lipschitz homeomorphic to closed balls. Assume that condition (1.2) holds, then BD p X q “ ℵ . Note that in the context of tilings, we say that two tilings are BD-equivalent if theircorresponding Delone sets, which are obtained by picking a point from each tile, are BD-equivalent. In addition to the above, [S2] contains an example of a substitution rule, forwhich the eigenvalues of the substitution matrix satisfy | λ | “ λ p d ´ q{ d (1.3)and the corresponding Delone sets are not uniformly spread, see [S2, Theorem 1.2 (III)].Note that in this example the main result of [S3] cannot be applied. Corollary 1.3.
There exists a primitive substitution tiling space X for which condition (1.3) holds and BD p X q “ ℵ . Cut-and-project sets:
Theorem 1.2 in [HKW] concerns the BD-equivalence relationin the context of cut-and-project sets that arise from linear toral flows (which constitutean equivalent method of constructing cut-and-project sets, see [ASW, Proposition 2.3]).Since the hull of a cut-and-project set is minimal, the corollary below follows directly from[HKW, Theorem 1.2 (III)] and our Theorem 1.1. We refer to [HKW] for more details onthe construction and terminology.
Corollary 1.4.
For almost every p k ´ d q -dimensional linear section S , which is a par-allelotope in the k -dimensional torus, there is a residual set of d -dimensional subspacesV for which the hull of the corresponding cut-and-project set contains continuously manydistinct BD-classes. In [FG, §
6] Frettl¨oh and Garber introduced the half-Fibonacci sets . These are cut-and-project sets in R that belong to the same hull, but due to their Theorem 6.4 arenot BD-equivalent to each other. In particular, they are not uniformly spread (see [FG,Theorem 3.2]). We thus obtain the following result. Corollary 1.5.
Let X be the hull of the half-Fibonacci sets from [FG] . Then BD p X q “ ℵ . Multiscale substitution tilings:
Multiscale substitution tilings were recently studiedin [SS]. Under an incommensurability assumption on the underlying substitution scheme,the corresponding tiling spaces are minimal and their associated Delone sets, which arenever FLC, are not uniformly spread. In view of this, our Theorem 1.1 implies thefollowing.
Corollary 1.6.
Let X be an incommensurable multiscale tiling space. Then BD p X q “ ℵ . Necessary and sufficient conditions for BD-non-equivalence
Notations.
Bold figures will be used to denote vectors in R d , and we will use thesupremum norm }¨} on R d throughout this document. Note that with respect to thisnorm, balls are (Euclidean) cubes, and we use both terms interchangeably. We denoteby B A, | A | and vol p A q the boundary, cardinality and Lebesgue measure of a set A Ă R d ,respectively, and we denote by S the cardinality of a finite set S . Given ε ą A Ă R d we denote the ε -neighborhood of A by A p` ε q def “ t x P R d | dist p x , A q ď ε u , D DICHOTOMY AND C-F CONVERGENCE 5 where dist p x , A q “ inf t} x ´ a } | a P A u . For an integer m ą Q d p m q def “ d ą i “ r a i , a i ` m q | a , . . . , a d P m Z + , the collection of all half-open cubes in R d with edge-length m and with vertices in m Z d ,and we denote by Q ˚ d p m q the collection of finite unions of elements from Q d p m q . In thecase m “ Q d and Q ˚ d . For A P Q d the notation vol d ´ pB A q standsfor the p d ´ q -Lebesgue measure of B A . The following lemma is a direct consequence ofLemmas 2.1 and 2.2 of [La]. Lemma 2.1.
Let F P Q ˚ d and let s ą , then vol ` pB F q p` s q ˘ ď c ¨ s d ¨ vol d ´ pB F q , (2.1) where c depends only on d . BD-equivalence.
The following condition for non-BD-equivalence of two Delonesets in R d was given in [FSS]. Theorem 2.2. [FSS, Theorem 1.1]
Let Λ , Λ be two Delone sets in R d and suppose thatthere is a sequence p A m q m P N of sets A m P Q ˚ d for which | p Λ X A m q ´ p Λ X A m q| vol d ´ pB A m q m Ñ8 ÝÝÝÑ 8 . Then there is no BD-map φ : Λ Ñ Λ . For the converse, we have the following result (compare [La, Lemma 2.3]).
Theorem 2.3.
Let Λ , Λ be two Delone sets in R d and suppose that there is no BD-mapbetween Λ and Λ . Then there is a sequence p A m q m P N of sets A m P Q ˚ d such that | p Λ X A m q ´ p Λ X A m q| vol d ´ pB A m q m Ñ8 ÝÝÝÑ 8 . (2.2) Proof.
Suppose that there is no BD-map between Λ and Λ , that is, no bijection φ :Λ Ñ Λ that satisfies sup x P Λ } x ´ φ p x q} ă 8 . For every m P N consider the bipartite graph G m def “ p Λ \ Λ , E m q , where E m “ ! t x , y u | x P Λ , y P Λ , } x ´ y } ď m ) . The existence of a perfect matching in G m for some m would imply the existence of aBD-map between Λ and Λ , contradicting our assumption. Thus by Hall’s marriagetheorem (see e.g. [Ra]), for every m P N there is a set X m Ă Λ i m , i m P t , u , so that X m ą p X p` m q m X Λ ´ i m q . Fix m P N , and assume without loss of generality that i m “
0. Set A m def “ ď t Q P Q d p m q | Q X X m ‰ ∅ u P Q ˚ d p m q . (2.3)For Q P Q d p m q let Q be a cube of edge-length 3 m which is concentric with Q , and set B m def “ ď t Q | Q P Q d p m q , Q X X m ‰ ∅ u P Q ˚ d p m q . YOTAM SMILANSKY AND YAAR SOLOMON
Clearly B m Ą A m Ą X m , and by the triangle inequality we have X p` m q m Ą B m . Therefore p Λ X A m q ą p Λ X B m q “ p Λ X A m q ` p Λ X p B m r A m qq , which implies p Λ X A m q ´ p Λ X A m q ą p Λ X p B m r A m qq . (2.4)It is left to show that p Λ Xp B m r A m qq{ vol d ´ pB A m q m Ñ8 ÝÝÝÑ 8 , which is a consequence ofthe following argument, taken from the proof of [La, Lemma 2.3]. We repeat Laczkovich’sreasoning for completeness.Denote by Q , . . . , Q n P Q d p m q the m -cubes whose union is A m and write B A m “ Ť sj “ F j , where each F j is a p d ´ q -dimensional face of some Q i j . For each 1 ď j ď s let P j P Q d p m q be the reflection of Q i j about F j . Then P j Ă B m r A m for every j , and in thesequence P , . . . , P s the same cube can appear at most 2 d times, the number of possiblereflections. Then2 d ¨ vol p B m r A m q ě s ÿ j “ vol p P j q “ s ¨ m d “ m ¨ s ¨ m d ´ “ m ¨ vol d ´ pB A m q . (2.5)Recall that Λ is a Delone set and let R P N be so that every cube of edge-length R contains a point of Λ . For simplicity, assume that m is an integer multiple of R .Consider the distinct elements of Q d p m q whose union is B m r A m . Each of these cubescontains at least p m { R q d points of Λ . Then p Λ X p B m r A m qq ě vol p B m r A m q R d ě m dR d vol d ´ pB A m q . (2.6)Combining (2.4) and (2.6) yields (2.2) with the sequence p A m q m P N defined by (2.3). (cid:3) Corollary 2.4.
Let p A m q m P N be a sequence of sets as in (2.2) , then for every R ą thereexists M ą so that for every m ě M each A m contains a ball of radius R .Proof. Let R ą m j Ñ 8 such that for every j theset A m j does not contain a ball of radius R . Then for every j we have A m j Ă pB A m j q p` R q and thus by Lemma 2.1 vol p A m j q ď c ¨ R d ¨ vol d ´ pB A m j q . (2.7)Since Λ and Λ are uniformly discrete and relatively dense, there exist constants a, b ą ja ¨ vol p A m j q ď p Λ X A m j q , p Λ X A m j q ď b ¨ vol p A m j q . (2.8)Combining (2.7) and (2.8) implies that for every j we have | p Λ X A m j q ´ p Λ X A m j q| vol d ´ pB A m j q ď p b ´ a q c ¨ R d , contradicting (2.2). (cid:3) D DICHOTOMY AND C-F CONVERGENCE 7 The topology on spaces of Delone sets
We consider the dynamical system p X, d, G q , where p X, d q is a compact metric spaceand G is a group acting on X . The dynamical system p X, d, G q is called minimal if every G -orbit , G.x def “ t g.x | g P G u for x P X , is dense in p X, d q . A set S Ă G is called syndetic if there is a compact set K Ă G so that for every g P G there is a k P K with kg P S .Note that when G “ R d this notion coincides with our definition of a relatively dense set.A point x P X is said to be uniformly recurrent if for every open neighborhood U of x the set of ‘return times’ to U , t g P G | g.x P U u , is syndetic. As shown in [Fu, Theorem1.15], in minimal systems every point is uniformly recurrent.Recall that given a metric ρ on R d we may use (1.1) to define a metric D on Cl p R d q , thespace of closed subsets of p R d , ρ q , and that this metric induces the Chabauty-Fell topology .Here and in what follows we take ρ to be the metric defined by the supremum norm }¨} on R d . Note that replacing it with any other norm on R d , such as the Euclidean norm,would change the metric D but not the induced Chabauty-Fell topology, also known as the local rubber topology in the context of aperiodic order. It is known that D is a completemetric on Cl p R d q , and the space ` Cl p R d q , D ˘ is compact, see e.g. [dH], [LSt].Let X be a collection of Delone sets in R d . Under the additional assumptions that X isa closed subset of Cl p R d q and that R d acts on X by translations, the space p X , D, R d q isa compact dynamical system. We say that Λ P X is almost repetitive if for every x P R d and ε ą R “ R p ε, x q ą B p y , R q in R d contains avector v P R d that satisfies D p Λ ´ x , Λ ´ v q ă ε. In words, for every x P R d and ε ą R ą B p , { ε qXp Λ ´ x q can be found in every R -ball, up to wiggling each point by at most ε . We also refer to[FR, Definitons 2.8, 2.13, 3.5] and to [LP] for distinctions between similar definitions ofrepetitivity.The observation in Lemma 3.1 is useful when working with the metric D in spaces ofuniformly discrete point sets. Lemma 3.1.
Suppose that Λ , Λ Ă R d are uniformly discrete sets with separation con-stant r ą , and that D p Λ , Λ q ă ε for ă ε ă r { . Then for every set A Ă B p , { ε q that is a translated copy of an element of Q ˚ d , there exist injective maps ϕ : Λ X A Ñ Λ X A p` ε q , ϕ : Λ X A Ñ Λ X A p` ε q , that satisfy @ x P Λ X A : } x ´ ϕ p x q} ă ε, @ y P Λ X A : } y ´ ϕ p y q} ă ε. (3.1) In particular, there is a constant c that depends on d and r so that | p Λ X A q ´ p Λ X A q| ď c ¨ ε d ¨ vol d ´ pB A q . (3.2) Proof.
Given A Ă B p , { ε q as above, since D p Λ , Λ q ă ε , the existence of ϕ , ϕ sat-isfying (3.1) follows directly from the definition of D in (1.1). Note that the maps areinjective since ε ă r {
2. Therefore | p Λ X A q ´ p Λ X A q| ď ` Λ X pB A q p` ε q ˘ ` ` Λ X pB A q p` ε q ˘ . Since Λ and Λ are uniformly discrete and in view of Lemma 2.1, (3.2) follows. (cid:3) YOTAM SMILANSKY AND YAAR SOLOMON
We remark that if Λ is a Delone set in R d with separation constant and packing radius r, R ą
0, and if X is the orbit closure of Λ with respect to D , then every Γ P X is a Deloneset with separation constant at least r and packing radius at most R .The following lemma shows that minimal spaces are uniformly almost repetitive . Namely,the radius R p x , ε q from the definition of almost repetitivity above does not depend on x . Lemma 3.2.
Let X be a compact space of Delone sets so that the dynamical system p X , D, R d q is minimal. Then for every ε ą there exists R “ R p ε q ą , so that for every Λ , Γ P X and y P R d , there exists some v P B p y , R q for which D p Γ , Λ ´ v q ă ε. Proof.
Let ε ą
0, and let Λ P X and x P R d . By minimality, the set Λ ´ x is uniformlyrecurrent. For η ą U x η def “ t Λ P X | D p Λ ´ x , Λ q ă η u , then the set t v P R d | Λ ´ v P U x ε { u is relatively dense (syndetic). In other words, there exists R x ε { ą R x ε { in R d contains some v P R d satisfying D p Λ ´ x , Λ ´ v q ă ε { t Λ ´ x | x P R d u is dense in X . Thus t U x ε { u x P R d is an open cover of X , and by compactness there exists a finite sub-cover U x ε { , . . . , U x n ε { .Then for every Γ P X these exists some j P t , . . . , n u so that Γ P U x j ε { , and hence D p Γ , Λ ´ x j q ă ε {
2. Setting R def “ max t R x ε { , . . . , R x n ε { u , it follows that for every y P R d there exists some v P B p y , R x j ε { q Ă B p y , R q such that D p Λ ´ x j , Λ ´ v q ă ε {
2. Then bythe triangle inequality D p Γ , Λ ´ v q ă ε , as required. (cid:3) A patch in a Delone set Λ is a finite subset of Λ. In the FLC setup it is often convenientto define specific elements in the orbit closure of Λ, also known as the hull, as nested unionsof patches. In such a case the limit set contains all the patches in the sequence. In contrastto the FLC case and in addition to other difficulties that arise in the non-FLC setup, thefact that two sets are close with respect to the metric D in (1.1) only gives informationabout behavior inside a large ball. In particular, convergence of a sequence of patches p Q m q m P N to a limit Γ does not imply that Γ contains any of the Q m ’s as sub-patches, oreven a patch that is close to any of the Q m ’s in the sense of Hausdorff distance. Theorem 3.3.
Let X be a minimal space of Delone sets in R d , Λ P X , p A m q m P N a sequenceof sets in Q ˚ d and p ε m q m ě a sequence of positive constants with ε ă r p Λ q , where r p Λ q isthe separation constant of Λ . Assume that the following properties hold for every m P N : (1) There exists x m P R d such that A m Ă B p x m , { ε m q . (2) There exists y m P R d such that B p y m , R p ε m ´ qq Ă A m , where R p ε q is as inLemma 3.2.Then there exist u m P B p y m , R p ε m ´ qq and patches Q m def “ p Λ X A m q ´ u m such that lim m Ñ8 Q m “ Γ P R d . Λ “ X . Moreover, for every m ě B p , R p ε m ´ qq Ă A m ´ u m Ă B p , { ε m q , (3.3) D p Λ ´ u m ´ , Λ ´ u m q ă ε m ´ , (3.4) D p Q m , Γ q ă ε m ´ (3.5) D DICHOTOMY AND C-F CONVERGENCE 9 and there exists c ą so that | p Γ X p A m ´ u m qq ´ Q m | ď c ¨ ε dm ¨ vol d ´ pB A m q , (3.6) where c depends on the dimension d and separation constant r p Λ q .Proof. First observe that by assumptions (1) and (2) we have 2 R p ε m q ď { ε m ` for every m P N . It follows from Lemma 3.2 that R p ε q ą { ε for every ε ą
0, and so we obtain ε m ` ď R p ε m q ď ε m . (3.7)In particular, the series ř m “ ε m is convergent.We define the vectors u m , and hence the patches Q m , inductively. ‚ By (1), A is in particular contained in a ball of radius 1 { ε . Let u be such that Q “ p Λ X A q ´ u is contained in B p , { ε q .Assume that the vectors u j , and thus the patches Q j “ p Λ X A j q ´ u j , are defined for j P t , . . . , m u such that for every 2 ď j ď m we have(i) B p , R p ε j ´ qq Ă A j ´ u j Ă B p , { ε j q .(ii) D p Λ ´ u j , Λ ´ u j ´ q ă ε j ´ .We define u m ` as follows. ‚ By (2), A m ` contains a ball of the form B p y m ` , R p ε m qq . By Lemma 3.2, let u m ` P B p y m ` , R p ε m qq be a vector satisfying D p Λ ´ u m , Λ ´ u m ` q ă ε m . (3.8)Thus (ii) for j “ m ` B p y m ` , R p ε m qq Ă A m ` and u m ` P B p y m ` , R p ε m qq we have B p , R p ε m qq Ă A m ` ´ u m ` . (3.9)By (1), A m ` ´ u m ` Ă B p x m ` ´ u m ` , { ε m ` q and by (3.9) A m ` ´ u m ` contains the origin. Then by the triangle inequality, A m ` ´ u m ` is contained in B p , { ε m ` q , completing the proof of (i) for j “ m ` u m and the patches Q m . Next we show thatthe sequence p Q m q m P N is a Cauchy sequence. Fix some ε ą M be so that 2 ε M ă ε .Let m ą n ą M , and note that by property (ii) we have D p Λ ´ u k ` , Λ ´ u k q ă ε k , forevery k ě M . Then by the triangle inequality, D p Λ ´ u m , Λ ´ u n q ď m ´ ÿ k “ n D p Λ ´ u k ` , Λ ´ u k q ă m ´ ÿ k “ n ε k ă ε n ă ε M ă ε, (3.10)where the third inequality follows from (3.7). By property (i), for every j P N the pointsets Q j and Λ ´ u j in particular coincide on the ball B p , { ε j ´ q . Since m, n ą M , thesets Λ ´ u n and Q n coincide on B p , { ε q , and similarly for Λ ´ u m and Q m . Therefore,relying on (3.10), for every m ą n ą M we have D p Q m , Q n q ď D ´ p Λ ´ u m q X B p , { ε q , p Λ ´ u n q X B p , { ε q ¯ ă ε. (3.11)Thus p Q m q m P N is a Cauchy sequence. The space p X , D q is complete, as a compact metricspace, hence the limit Γ def “ lim m Ñ8 Q m “ lim m Ñ8 Λ ´ u m exists and belongs to X .It is left to prove (3.3), (3.4), (3.5) and (3.6). First observe that (3.3) and (3.4) followimmediately from the construction, see properties (i) and (ii). To see (3.5), let m P N . Let k ą m be so that D p Q k , Γ q ă ε m . Repeating the computations in (3.10) and (3.11)yields that D p Q m , Q k q ă ε m , and by (3.7) we have D p Q m , Γ q ď D p Q m , Q k q ` D p Q k , Γ q ă ε m ă ε m ´ . Finally, we prove (3.6). By (3.3) we have A m ´ u m Ă B p , { ε m q and by (3.5) we have D p Γ , Q m ` q ă ε m . Thus by Lemma 3.1 with A “ A m ´ u m we obtain | p Γ X p A m ´ u m qq ´ p Q m ` X p A m ´ u m qq| ď c ¨ ε dm ¨ vol d ´ pB A m q . (3.12)By (3.4) we have D p Λ ´ u m , Λ ´ u m ` q ă ε m , and applying Lemma 3.1 once again we get | pp Λ ´ u m ` q X p A m ´ u m qq ´ pp Λ ´ u m q X p A m ´ u m qq| ď c ¨ ε dm ¨ vol d ´ pB A m q . By the definition of the Q m ’s, and since A m ´ u m Ă A m ` ´ u m ` by (3.3), this is exactly | p Q m ` X p A m ´ u m qq ´ Q m | ď c ¨ ε dm ¨ vol d ´ pB A m q . (3.13)Combining (3.12) and (3.13) yields (3.6) and completes the proof of the theorem. (cid:3) Finding patches with large discrepancy
The goal of this chapter is to prove the following proposition, which will be used in ourproof of Theorem 1.1 in § Proposition 4.1.
Let Λ Ă R d be a non-uniformly spread Delone set. Then there exist asequence p A m q m P N of sets in Q ˚ d and a sequence p x m q m P N of vectors in R d so that | p Λ X A m q ´ p Λ X p A m ` x m qq| vol d ´ pB A m q m Ñ8 ÝÝÝÑ 8 . (4.1)Let Λ Ă R d be a Delone set. We define the central lower density and the central upperdensity of Λ respectively by∆ ˚ p Λ q def “ lim inf t Ñ8 p B p , t q X Λ q vol p B p , t qq ∆ ˚ p Λ q def “ lim sup t Ñ8 p B p , t q X Λ q vol p B p , t qq . If the limit lim t Ñ8 p B p , t q X Λ q { vol p B p , t qq exists, it is called the central density of Λand is denoted by ∆ p Λ q .We begin with the following lemma. Lemma 4.2.
Let Λ be a Delone set, γ ą and A P Q ˚ d . Then for every ε ą there exists K ą such that for every integer k ě K : (1) if p Λ X B p ,k qq vol p B p ,k qq ě γ then the ball B p , k q contains A ` x , a translated copy of A with x P Z d , such that p Λ X p A ` x qq vol p A q ě γ ´ ε. (2) if p Λ X B p ,k qq vol p B p ,k qq ď γ then the ball B p , k q contains A ` x , a translated copy of A with x P Z d , such that p Λ X p A ` x qq vol p A q ď γ ` ε. D DICHOTOMY AND C-F CONVERGENCE 11
Proof.
This is a simple averaging argument. We prove property (1), the proof of property(2) is similar.Denote by ρ the diameter of the set A . For a large integer k we write B p , k q “ B r´ ρ s \ pB B q r` ρ s , where B r´ ρ s , pB B q r` ρ s P Q ˚ d are defined by B r´ ρ s def “ ď t Q P Q d | Q Ă B p , k q , dist p Q, B B p , k qq ą ρ upB B q r` ρ s def “ B p , k q r B r´ ρ s , (4.2)where dist p X, Y q def “ inf t} x ´ y } | x P X, y P Y u .Given ε ą K P N large enough so that for every integer k ě K we havevol ` pB B q r` ρ s ˘ vol p B p , k qq ă ε . (4.3)Let k ě K such that p Λ X B p , k qq vol p B p , k qq ě γ, (4.4)and let N k def “ t x P Z d | A ` x Ă B p , k qu . By way of contradiction, assume that @ x P N k : p Λ X p A ` x qq ă p γ ´ ε q vol p A q . (4.5)Notice that the number of cubes from Q d that form A is vol p A q . Then by countingthe points of Λ (with multiplicity) in all the sets A ` x , x P N k , the points in every unitlattice cube in B r´ ρ s is counted exactly vol p A q times. Thus N k p γ ´ ε q vol p A q (4.5) ą ÿ x P N k p Λ X p A ` x qq ě vol p A q ¨ ` Λ X B r´ ρ s ˘ . (4.6)Note that N k ď vol p B p , k qq , then dividing both sides of (4.6) by vol p A q ¨ vol p B p , k qq yields γ ´ ε ą ` Λ X B r´ ρ s ˘ vol p B p , k qq (4.2) ě p Λ X B p , k qq vol p B p , k qq ´ ` Λ X pB B q r` ρ s ˘ vol p B p , k qq (4.3) , (4.4) ą γ ´ ε , a contradiction. (cid:3) Lemma 4.3.
Suppose that Λ is a Delone set in R d and that ∆ ˚ p Λ q ă ∆ ˚ p Λ q . Then thereexist α ă β , integers a k Ñ 8 and x k P Z d such that p Λ X B p , a k qq vol p B p , a k qq ď α and p Λ X B p x k , a k qq vol p B p x k , a k qq ě β. Proof.
By the assumption on the densities, there exist sequences a k , b l Ñ 8 so thatlim k Ñ8 p Λ X B p , a k qq vol p B p , a k qq “ ˜ α and lim l Ñ8 p Λ X B p , b l qq vol p B p , b l qq “ ˜ β, where ˜ α ă ˜ β . Since Λ is uniformly discrete, and since the p d ´ q -volume of the boundaryof a cube grows slower than the cube’s volume, we may assume that the numbers a k , b k are integers. Let δ ă ˜ β ´ ˜ α and fix K P N such that for every k, l ě K we have p Λ X B p , a k qq vol p B p , a k qq ď ˜ α ` δ and p Λ X B p , b l qq vol p B p , b l qq ě ˜ β ´ δ. (4.7) For every k , applying Lemma 4.2 with A “ B p , a k q , ε “ ˜ β ´ ˜ α ´ δ ą
0, and ˜ β ´ δ in therole of γ , and combining this with (4.7), we find a large enough l “ l k and x k P Z d so that B p , b l q contains the ball B p x k , a k q , which satisfies p Λ X B p x k , a k qq vol p B p x k , a k qq ě p ˜ β ´ δ q ´ ε “ ˜ β ´ ˜ β ´ ˜ α . (4.8)Setting α def “ ˜ α ` ˜ β ´ ˜ α and β def “ ˜ β ´ ˜ β ´ ˜ α , the assertion follows from (4.7) and (4.8). (cid:3) Proof of Proposition 4.1.
Let Λ Ă R d be a non-uniformly spread Delone set. In view ofLemma 4.3 we may further assume that ∆ def “ ∆ p Λ q exists. For α ‰ ∆ ´ { d the Delonesets α Z d and Λ do not have the same central density and hence there is no BD-mapbetween them (see e.g. [FSS, Corollary 3.2]). By our assumption on Λ, there is no BD-map between Λ and ∆ ´ { d Z d as well. Applying Theorem 2.3 on these two Delone sets weobtain a sequence p A m q m P N of sets in Q ˚ d that satisfies ˇˇ p ∆ ´ { d Z d X A m q ´ p Λ X A m q ˇˇ vol d ´ pB A m q m Ñ8 ÝÝÝÑ 8 . By passing to a subsequence of p A m q m P N we may assume that p ∆ ´ { d Z d X A m q ´ p Λ X A m q vol d ´ pB A m q m Ñ8 ÝÝÝÑ 8 , (4.9)and complete the proof using (1) of Lemma 4.2. In the case that p ∆ ´ { d Z d X A m q ă p Λ X A m q for all large values of m , the proof is similar using (2) of Lemma 4.2 insteadof (1).For every m P N we pick ε m such that ε m vol p A m q ă vol d ´ pB A m q (4.10)and apply Lemma 4.2 with γ “ ∆ ´ ε m , A “ A m and ε “ ε m . Note that since ∆ p Λ q “ ∆exists, the condition p Λ X B p ,k qq vol p B p ,k qq ě ∆ ´ ε m is satisfied for any sufficiently large k . By (1)of Lemma 4.2, in particular, there exists some K m P N and a vector x m P Z d so that A m ` x m Ă B p , K m q and p Λ X p A m ` x m qq vol p A m q ě ∆ ´ ε m . (4.11)By (4.9) p ∆ ´ { d Z d X A m q ´ p Λ X p A m ` x m qqq vol d ´ pB A m q ` p Λ X p A m ` x m qq ´ p Λ X A m q vol d ´ pB A m q m Ñ8 ÝÝÝÑ 8 . (4.12)Note that p ∆ ´ { d Z d X A m q ď ∆ ¨ vol p A m q ` c ¨ vol d ´ pB A m q , where c depends on d and ∆, and by (4.11) we also have p Λ X p A m ` x m qq ě p ∆ ´ ε m q vol p A m q . Then p ∆ ´ { d Z d X A m q ´ p Λ X p A m ` x m qqq ď c ¨ vol d ´ pB A m q ` ε m vol p A m q (4.10) ď c ¨ vol d ´ pB A m q , (4.13) D DICHOTOMY AND C-F CONVERGENCE 13 where c depends on d and ∆. Thus plugging (4.13) in (4.12) completes the proof ofProposition 4.1. (cid:3) Proof of Theorem 1.1
This chapter contains the proof of Theorem 1.1. Let Λ Ă R d be a non-uniformly spreadDelone set and let A m P Q ˚ d and x m P Z d be the sets and vectors that were constructedin Proposition 4.1. Let ε m ą A m is contained in a ball of radius 1 { ε m . Itfollows from Corollary 2.4 that equation (4.1) implies that for every R there exists M sothat A m contains a ball of radius R for every m ě M . Then by passing to subsequences p A m j q j P N , p x m j q j P N and p ε m j q j P N , we may assume that the set A m j contains a ball of radius2 R p ε m j ´ q . To simplify notations we keep using the lower index m for this new sequence,and so we have a sequence p A m q m P N that satisfies Proposition 4.1, and y m , z m P R d forwhich B p y m , R p ε m ´ qq Ă A m Ă B p z m , { ε m q . (5.1)Denote B m def “ A m ` x m , p m def “ y m ` x m , q m def “ z m ` x m . (5.2)Then B p p m , R p ε m ´ qq Ă B m Ă B p q m , { ε m q . (5.3)By (4.1), there is a sequence of constants µ m Ñ 8 such that | p Λ X A m q ´ p Λ X p A m ` x m qq| “ µ m ¨ vol d ´ pB A m q . (5.4)Since µ m Ñ 8 , by passing to a further subsequence mutually for A m , x m , ε m and µ m ,we may assume that µ m approaches infinity in an extremely fast rate. In particular, bydefining every element in the sequence with dependence on the previous one, we mayassume that R p ε m ´ q d µ m m Ñ8 ÝÝÝÑ . (5.5)Using these notations, Theorem 1.1 follows from Lemmas 5.1 and 5.2 below. Lemma 5.1.
Let X be a minimal space of Delone sets and assume that there exists Λ P X that is non-uniformly spread. Let p A m q m P N and p B m q m P N be the sequences of sets in Q ˚ d defined in Proposition 4.1 and in (5.2) , with respect to Λ . For every word ω P t
A, B u N let p C m q m P N be the sequence of sets in Q ˚ d defined by C m def “ A m , ω p m q “ AB m , ω p m q “ B, (5.6) where w p m q is m ’th letter in w . Then there exists a sequence p u m q m P N of vectors in R d so that Λ ω “ lim m Ñ8 p Λ X C m q ´ u m is a Delone set in X , u m P B p y m , R p ε m ´ qq , ω p m q “ AB p p m , R p ε m ´ qq , ω p m q “ B, (5.7) and @ m ě | p Λ ω X p C m ´ u m qq ´ p Λ X C m q| ď c ¨ vol d ´ pB C m q , (5.8) where c is a constant that depends on d and on the separation constant r p Λ q . Proof.
Given ω P t
A, B u N , consider the sequence p C m q m P N of sets in Q ˚ d defined by (5.6).By (5.1) and (5.3), conditions (1) and (2) of Theorem 3.3 are being satisfied for p C m q m P N ,with p ε m q m P N as described at the beginning of this section. Applying Theorem 3.3 weobtain vectors u m satisfying (5.7), for which the sequence of patches Q m def “ p Λ X C m q ´ u m is convergent. Setting Λ ω to be the limit set, it is left to obtain (5.8). Indeed, by (3.6) ofTheorem 3.3, for every m ě | p Λ ω X p C m ´ u m qq ´ Q m | ď c ¨ ε dm ¨ vol d ´ pB C m q , (5.9)where c depends on d and on r p Λ q . Clearly Q m “ p Λ X C m q , thus (5.9) implies (5.8)and the proof is complete. (cid:3) Lemma 5.2.
Let X be a minimal space of Delone sets and assume that there exists Λ P X that is non-uniformly spread. Let η, σ P t
A, B u N be two words that differ in infinitelymany places. Then the Delone sets Λ η and Λ σ , which are defined in Lemma 5.1, areBD-non-equivalent.Proof. Let η, σ
P t
A, B u N be two sequences that differ in infinitely many places. Forsimplicity, we assume that η p m q ‰ σ p m q for every m P N , which can be achieved bytaking a subsequence of the indices, entailing no loss of generality to the remainder of theproof. We use an upper index of η or σ on elements of Q ˚ d and on vectors, e.g. C ηm and u σm , to distinguish between those elements that come from the construction of Λ η and ofΛ σ in Lemma 5.1.Let r u ηm P Z d be an integer vector which is closest to u ηm , then F m def “ C ηm ´ r u ηm is anelement of Q ˚ d and | p Λ η X F m q ´ p Λ η X p C ηm ´ u ηm qq| ď c ¨ vol d ´ B F m , where by (2.1), c is a constant depending on d and on r p Λ q . Combining this with (5.8)for w “ η we obtain that @ m ě | p Λ η X F m q ´ p Λ X C ηm q| ď p c ` c q vol d ´ B F m (5.10)Next observe that for every m ě v m P R d so that p C ηm ´ u ηm q ´ v m “ C σm ´ u σm and } v m } ď R p ε m ´ q (5.11)Indeed, assume without loss of generality that η p m q “ A and σ p m q “ B , then combining(5.2), (5.6) and (5.7) yields that C ηm “ A m , C σm “ A m ` x m , u ηm P B p y m , R p ε m ´ qq and u σm P B p y m ` x m , R p ε m ´ qq , which implies (5.11). By (5.11), the symmetric difference of C σm ´ u σm and F m satisfies @ m ě p C σm ´ u σm q △ F m Ă B F p` c ¨ R p ε m ´ qq m , and hence by (2.1) @ m ě | p Λ σ X F m q ´ p Λ σ X p C σm ´ u σm qq| ď c ¨ R p ε m ´ q d ¨ vol d ´ B F m , where c , c depends on d and on r p Λ q . Again by (5.8), this time with w “ σ , we obtain @ m ě | p Λ σ X F m q ´ p Λ X C σm q| ď ` c ` c ¨ R p ε m ´ q d ˘ vol d ´ B F m . (5.12) D DICHOTOMY AND C-F CONVERGENCE 15
In view of (5.10), (5.12) and the triangle inequality, for every m ě | p Λ η X F m q ´ p Λ σ X F m q| ě| p Λ X C ηm q ´ p Λ X C σm q| ´ | p Λ η X F m q ´ p Λ X C ηm q| ´ | p Λ σ X F m q ´ p Λ X C σm q| ě| p Λ X C ηm q ´ p Λ X C σm q| ´ c ¨ R p ε m ´ q d ¨ vol d ´ pB F m q , (5.13)where c depends on d and r p Λ q . Since C ηm “ A m , C σm “ A m ` x m and vol d ´ pB A m q “ vol d ´ pB F m q , by (5.4) we have | p Λ X C ηm q ´ p Λ X C σm q| “ µ m ¨ vol d ´ pB F m q . (5.14)Combining (5.13) and (5.14) we see that | p Λ η X F m q ´ p Λ σ X F m q| ě ` µ m ´ c ¨ R p ε m ´ q d ˘ vol d ´ pB F m q“ ˆ µ m ˆ ´ c ¨ R p ε m ´ q d µ m ˙˙ vol d ´ pB F m q . (5.15)Therefore, by (5.4), (5.5) and (5.15) we obtain | p Λ η X F m q ´ p Λ σ X F m q| vol d ´ pB F m q ě ˆ µ m ˆ ´ c ¨ R p ε m ´ q d µ m ˙˙ m Ñ8 ÝÝÝÑ 8 . Theorem 2.2 then implies that the Delone sets Λ η and Λ σ are BD-non-equivalent, asrequired. (cid:3) Given Lemma 5.2, Theorem 1.1 follows.
Proof of Theorem 1.1.
Let X be a minimal space of Delone sets. If there exists a uniformlyspread Λ P X , then as noted in § P X is uniformly spread, and (1) holds.Otherwise, there exists some Λ P X that is non-uniformly spread. Consider the equiva-lence relation on t A, B u N in which η „ σ if η and σ differ in only finitely many places, andlet Ω Ă t
A, B u N be a set of equivalence class representatives. Since every equivalence classin this relation is countable, | Ω | “ ℵ . For every two distinct words η, σ P Ω, Lemma 5.2implies that Λ η and Λ σ are BD-non-equivalent, therefore BD p X q ě ℵ . As explained in § (cid:3) References [ASW] F. Adiceam, Y. Solomon, B. Weiss,
Cut-and-project quasicrystals, lattices, and dense forests ,preprint. https://arxiv.org/abs/1907.03501.[ACG] J. Aliste-Prieto, D. Coronel, J. M. Gambaudo,
Linearly repetitive Delone sets are rectifiable , Ann.Inst. H. Poincar´e Anal. Non Lin´eaire 30 (2), 275-290, (2013).[BG] M. Baake, U. Grimm, Aperiodic order. Volume 1: A mathematical invitation, Cambridge Univer-sity Press, Cambridge (2013).[BK] D. Burago, B. Kleiner,
Separated nets in Euclidean space and Jacobians of biLipschitz maps , Geom.Func. Anal. 8, no.2, 273-282, (1998).[Ch] C. Chabauty,
Limite d’ensembles et g´eom´etrie des nombres , Bull. Soc. Math. France, 78, 143-151,(1950).[dH] P. de la Harpe,
Spaces of closed subgroups of locally compact groups , preprint, arXiv:0807.2030,(2008).[DSS] W. A. Deuber, M. Simonovits, V. T. S´os,
A note on paradoxical metric spaces , Studia Sci. Math.Hungar 30 (1), 17-24, (1995).[DO1] M. Duneau, C. Oguey,
Displacive transformations and quasicrystalline symmetries , J. Phys. 51(1), 5-19, (1990).[DO2] M. Duneau, C. Oguey,
Bounded interpolation between lattices , J. Phys. A 24, 461-475, (1991). [Fe] J.M. Fell.
A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space , Proc.Amer. Math. Soc. 13(3), 472-476, (1962).[FG] D. Frettl¨oh, A. Garber,
Pisot substitution sequences, one dimensional cut-and-project sets andbounded diameter sets with fractal boundary , Indag. Math. (N.S.) 29 (4), 1114-1130, (2018).[FR] D. Frettl¨oh, C. Richards,
Dynamical properties of almost repetitive Delone sets , Disc. Cont. Dyna.Syst. 34 (2), 531-556, (2014).[FSS] D. Frettl¨oh, Y. Smilansky, Y. Solomon,
Bounded displacement non-equivalence in substitutiontilings , J. Comb. Thoery, Series A. 177, (2021). https://doi.org/10.1016/j.jcta.2020.105326.[Fu] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton Univ.Press, Princeton, (1981).[HKK] A. Haynes, M. Kelly, H. Koivusalo,
Constructing bounded remainder sets and cut-and-project setswhich are bounded distance to lattice, II , Indag. Math. (N.S.) 28 (1), 138-144, (2017).[HKW] A. Haynes, M. Kelly, B. Weiss,
Equivalence relations on separated nets arising from linear toralflows , Proc. Lond. Math. Soc. 109 (5), 1203-1228, (2014).[HK] A. Haynes, H. Koivusalo,
Constructing bounded remainder sets and cut-and-project sets which arebounded distance to lattice , Israel J. Math. 212, 189-201, (2016).[KS1] M. Kelly, L. Sadun,
Pattern equivariant cohomology and theorems of Kesten and Oren , Bull. Lond.Math. Soc. 47 (1), 13-20, (2015).[KS2] M. Kelly, L. Sadun,
Pattern equivariant mass transport in aperiodic tilings and cohomology , Int.Math. Res. Not. IMRN, (2018).[La] M. Laczkovich,
Uniformly spread discrete sets in R d , J. Lond. Math. Soc. 46 (2), 39-57, (1992).[LP] J. C. Lagarias, P. A. B. Pleasants, Repetitive Delone sets and quasicrystals , Ergo. Theo. Dyna. Sys.23, 831-867, (2003).[LSt] D. Lenz, P. Stollmann,
Delone dynamical systems and associated random operators , in J. M.Combes, J. Cuntz, G. A. Elliott, G. Nenciu, H. Siedentop and S. Stratila (Eds.), Operator alge-bras and mathematical physics (Constanta 2001), Theta, Bucharest, 267-285, (2003).[Mag] A. N. Magazinov,
The family of bi-Lipschitz classes of Delone sets in Euclidean space has thecardinality of the continuum , Proc. of the Steklov. Inst. of Math. 275, 87-98, (2011).[McM] C. T. McMullen,
Lipschitz maps and nets in Euclidean space , Geom. Func. Anal. 8, no.2, 304-314,(1998).[Ra] R. Rado.
Factorization of even graphs , Q. J. Math. 1, 95-104, (1949).[SS] Y. Smilansky, Y. Solomon,
Multiscale substitution tilings , arXiv:2003.11735, (2020).[S1] Y. Solomon,
Substitution tilings and separated nets with similarities to the integer lattice , Israel J.Math. 181, 445-460, (2011).[S2] Y. Solomon,
A simple condition for bounded displacement , J. Math. Anal. Appl. 414 (1), 134-148,(2014).[S3] Y. Solomon,
Continuously many bounded displacement non-equivalences in substitution tiling spaces ,J. Math. Anal. Appl. 492 (1), (2020). https://doi.org/10.1016/j.jmaa.2020.124426.
Yotam SmilanskyDepartment of Mathematics, Rutgers University, NJ, USA. [email protected]