A dichotomy for bounded displacement equivalence of Delone sets

We prove that in every compact space of Delone sets in $\mathbb{R}^d$ which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty--Fell topology, which is the natural topology on the space of closed subsets of $\mathbb{R}^d$. This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.


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 0 so that every ball of radius r contains at most one point of Λ and Λ intersects every ball of radius R. We refer to r and R as the separation constant and the packing radius of Λ, respectively. Two Delone sets Λ, Γ Ă R d are said to be bounded displacement (BD) equivalent if there exists a bijection φ : Λ Ñ Γ satisfying sup xPΛ }x´φpxq} ă 8.
Such a mapping φ is called a BD-map. Note that since norms in R d are equivalent, this definition does not depend on the choice of norm. Lattices in R d with the same covolume are BD-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 αZ d , for some α ą 0.
Fix a metric ρ on R d and consider the space C pR d q of closed subsets of R d . The Chabauty-Fell topology on C pR d q is the topology induced by the metric (see Appendix A) where Bpx, Rq is the open ball of radius R ą 0 centered at x P R d with respect to the metric ρ, and A p`εq is the ε neighborhood of the set A. In this work we only consider metrics ρ that are determined by norms on R d , and although different choices of norms result in different metrics D, they all define the same topology. We remark that in the aperiodic order literature, this topology, which was introduced by Chabauty [Ch] for C pR d q as well as for a more general setting, and later extended by Fell [Fe], is often referred to as the natural topology or the local rubber topology, see e.g. [BG,§5]. See also [LSt] for the relation to the Hausdorff metric. Delone sets in R d are elements of C pR d q, and we may consider compact spaces of Delone sets, where the implied limits are taken with respect to the Chabauty-Fell topology. Such a space X of Delone sets in R d is minimal with respect to the R d action by translations if 1 the orbit closure of every Delone set Λ P X is dense in X. Minimality of X is equivalent to the recurrence of patches in each Delone set Λ P X, where a patch is a finite subset of a Delone set. This important geometric consequence of minimality is called almost repetitivity, and a precise definition and additional details will be given in §3. For a proof of this equivalence see [FR,Theorem 3.11] and [SS,Theorem 6.5], and see also the discussion included in [KL].
Denote the cardinality of the set of BD-equivalence classes represented in X by BDpXq. 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 with respect to the Chabauty-Fell topology and minimal with respect to the action of R d by translations. Then either (1) there exists a uniformly spread Delone set in X (and so every Λ P X is uniformly spread and BDpXq " 1), or (2) BDpXq " 2 ℵ 0 , where 2 ℵ 0 denotes the cardinality of the continuum.
Observe that the minimality assumption is essential, as shown by the following simple example. Consider Λ " p´2Nq \ t0u \ 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 of translations of Λ, the orbit closure of Z and the orbit closure of 2Z. Therefore BDpXq " 3, and indeed X is not minimal.
Let us describe the proof of Theorem 1.1. The implication in the brackets of (1) is a direct consequence of [La,Theorem 1.1], see also [FG,Theorem 3.2] for a sketch of a similar proof that holds for general minimal spaces of Delone sets. A uniformly discrete set in R d with separation constant r ą 0 is BD-equivalent to a subset of the lattice r 2 Z d , hence the upper bound BDpXq ď 2 ℵ 0 is trivial. We prove the remaining implication according to the following steps. Given a non-uniformly spread Delone set in a minimal space X, we construct in §4 a sequence of pairs of patches consisting of an increasingly deviant number of points. by §3, choosing a patch from each pair, which corresponds to the choice of a word on a two letter alphabet, gives rise to a Delone sets in X with certain properties. Finally, it is shown in §5 that using the equivalent condition for non-BD equivalence of two Delone sets established in §2, two Delone sets defined using words that differ in infinitely many places are BD-non-equivalent, and so X contains continuously many BD-equivalence classes. y many places must be BD-non-equivalent.
Recall that a Delone set Λ has finite local complexity (FLC) if for every R ą 0 the number of distinct patterns that are contained in balls of radius R in Λ up to translations is finite. In such case every Delone set in the orbit closure of Λ under translations, sometimes called the hull of Λ, also has FLC. The hull itself is then called FLC, and the Chabauty-Fell topology on X coincides with the local topology, see [BG,§5]. It follows that Theorem 1.1 holds also for FLC spaces with respect to the local topology, and constitutes a new result both in the FLC and non-FLC setup. In particular, it answers question (1) in [FG,§7] in the strongest possible way.
In addition to Theorem 1.1, we establish in Theorem 2.3 a useful equivalent condition for two Delone sets to be non-BD equivalent. This result is the converse of the implication of Theorem 2.2 which first appeared in [FSS], and may be of interest in its own right.
Delone sets are mathematical models of atomic positions, and BD-equivalence offers a natural way of classifying them. BD-equivalence for general discrete point sets was previously considered mainly in the context of uniformly spread point sets, see e.g. [DO1,DO2,La] and [DSS]. In recent years, BD-equivalence emerged as an object of study for Delone sets that appear in the study of mathematical quasicrystals and aperiodic order, see [BG] for a comprehensive introduction to such constructions. For cut-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 be uniformly spread.
Recently, questions regarding BD-non-equivalence between two Delone sets were considered in [FSS], where a sufficient condition for BD-non-equivalence was established. It was later shown in [S3] that if the eigenvalues and eigenspaces of the substitution matrix satisfy a certain condition, then the corresponding substitution tiling space contains continuously many distinct BD-classes.
The following less restrictive equivalence relation on Delone sets is often studied in parallel to the BD-equivalence relation. We say that two Delone sets Λ and Γ are biLipschitz (BL) equivalent if there exists a biLipschitz bijection between them. Namely, a bijection ϕ : Λ Ñ Γ and a constant C ě 1 so that It was shown by Burago and Kleiner [BK1] and independently by McMullen [McM], that there exist Delone sets in R d , d ě 2, that are not BL-equivalent to a lattice in R d . It was shown in [Mag] that there are continuously many Delone sets that are pairwise BL-nonequivalent, and a hierarchy of equivalence relations on Delone sets, which includes BD and BL equivalence, was recently introduced in [DK]. It would be interesting to obtain an analogue of our Theorem 1.1 in this context.
Question. Does Theorem 1.1 hold if BD-equivalence is replaced by BL-equivalence?
In view of the sufficient condition for BL-equivalence to a lattice given by Burago and Kleiner in [BK2] and the constructions in [CN] and [Mag], we remark that the results given in §3 and §4 regarding densities and discrepancy estimates may be relevant also in the study of BL-non-equivalence and the question stated above.
1.1. Consequences of Theorem 1.1. Theorem 1.1 directly implies that BDpXq " 2 ℵ 0 for many special families of minimal spaces of Delone sets which are central in the theory of 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 1 ą |λ 2 | ě . . . ě |λ n | the eigenvalues of the substitution matrix, and we let t ě 2 be the minimal index such that the eigenspace of λ t contains non-zero vectors whose sum of coordinates is not zero. Under the assumption that tiles are bi-Lipschitz homeomorphic to closed balls, it was shown in [S2,Theorem 1.2 (I)] that if |λ t | ą λ pd´1q{d 1 (1.2) then the Delone sets corresponding to the tilings in the tiling space are not uniformly spread. Under the assumption (1.2) and an additional assumption regarding the existence of certain patches, it was recently shown in [S3] that BDpXq " 2 ℵ 0 . Given the above result of [S2], and since substitution tiling spaces are minimal (see [BG]), the following strengthening of the main result of [S3] is a direct consequence of our Theorem 1.1.
Corollary 1.2. Let X be a primitive substitution tiling space with tilings by tiles that are bi-Lipschitz homeomorphic to closed balls. Assume that condition (1.2) holds, then BDpXq " 2 ℵ 0 .
Note that in the context of tilings, we say that two tilings are BD-equivalent if their corresponding Delone sets, which are obtained by picking a point from each tile, are BDequivalent. In addition to the above, [S2] contains an example of a substitution rule, for which the eigenvalues of the substitution matrix satisfy 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 BDpXq " 2 ℵ 0 .
1.1.2. Cut-and-project sets: Theorem 1.2 in [HKW] concerns the BD-equivalence relation in the context of cut-and-project sets that arise from linear toral flows (which constitute an 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 on the construction and terminology.
Corollary 1.4. For almost every pk´dq-dimensional linear section S, which is a parallelotope in the k-dimensional torus, there is a residual set of d-dimensional subspaces V for which the hull of the corresponding cut-and-project set contains continuously many distinct BD-classes.
The half-Fibonacci sets were introduced in [FG,§6]. These are cut-and-project sets in R that belong to the same hull and are BD-non-equivalent. In particular, they are not uniformly spread (see [FG,Theorem 3.2]). We thus obtain the following result.
1.1.3. Multiscale substitution tilings: Multiscale substitution tilings were recently studied in [SS]. Under an incommensurability assumption on the underlying substitution scheme the corresponding tiling spaces are minimal [SS,§6], and combined with a mild assumption on the boundaries of the prototiles which holds for example for polygonal tiles, their associated Delone sets, which are never FLC, are also never uniformly spread [SS,§8].
A proof of Theorem 1.1 in the FLC setup was given in [FGS], which appeared after the first version of this paper came out. Their work is independent of ours.

Necessary and sufficient conditions for BD-non-equivalence
2.1. Notations. Bold figures will be used to denote vectors in R d , and we will use the supremum norm }¨} 8 on R d throughout this document. Note that with respect to this norm, balls are (Euclidean) cubes, and we use both terms interchangeably. We denote by BA, |A| and volpAq 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 0 and For an integer m ą 0 we denote by the collection of all half-open cubes in R d with edge-length m and with vertices in mZ d , and we denote by Qdpmq the collection of finite unions of elements from Q d pmq. In the case m " 1 we simply write Q d and Qd. For A P Q d the notation vol d´1 pBAq stands for the pd´1q-Lebesgue measure of BA. The following lemma is a direct consequence of Lemmas 2.1 and 2.2 of [La].
Lemma 2.1. Let F be a translated copy of an element of Qd and let s ą 0, then where c 0 depends only on d.
2.2. BD-equivalence. The following condition for non-BD-equivalence of two Delone sets in R d was given in [FSS].
Theorem 2.2. [FSS, Theorem 1.1] Let Λ 0 , Λ 1 be two Delone sets in R d and suppose that there is a sequence pA m q mPN of sets, A m P Qd, for which Then there is no BD-map φ : Λ 0 Ñ Λ 1 .
We show that the converse also holds (compare [La,Lemma 2.3]).
Theorem 2.3. The following are equivalent for two Delone sets There is a sequence pA m q mPN of sets, which are translated copies of elements of Qd, such that Proof. The implication piiq ñ piq follows from Theorem 2.2, since translating the sets A m by at most ? d changes the numerator by at most a constant times vol d´1 pBA m q. For piq ñ piiq, suppose that there is no BD-map between Λ 0 and Λ 1 , that is, no bijection φ : Λ 0 Ñ Λ 1 that satisfies sup ) .
The existence of a perfect matching in G m for some m would imply the existence of a BD-map between Λ 0 and Λ 1 , contradicting our assumption. Thus by Hall's marriage theorem (see e.g. [Ra]), for every m P N there is a set X m Ă Λ im , i m P t0, 1u, so that #X m ą #pX p`2mq m X Λ 1´im q. Fix m P N, and assume without loss of generality that For Q P Q d pmq let Q 1 be a cube of edge-length 3m which is concentric with Q, and set It is left to show that #pΛ 1 XpB m A m qq{vol d´1 pBA m q mÑ8 ÝÝÝÑ 8, which is a consequence of the following argument, taken from the proof of [La,Lemma 2.3]. Suppose that BA m consists of s faces of cubes in Q d pmq. For each such face, let P j be the cube in Q d pmq contained in B m A m with boundary containing that face. Note that P 1 , . . . , P s are not necessarily distinct and that each cube has 2d faces, and so 2d¨volpB m A m q ě s ÿ j"1 volpP j q " s¨m d " m¨s¨m d´1 " m¨vol d´1 pBA m q.
The relative denseness of Λ 1 implies that #pΛ 1 X pB m A m qq ě c¨volpB m A m q for some constant c ą 0 independent of m, and the proof follows.
Corollary 2.4. Let pA m q mPN be a sequence of sets as in (2.2), then for every R ą 0 there exists M ą 0 so that for every m ě M each A m contains a ball of radius R.
Proof. Let R ą 0 and suppose that there is a sequence m j Ñ 8 such that for every j the set A m j does not contain a ball of radius R. Then for every j we have A m j Ă pBA m j q p`Rq and thus by Lemma 2.1 volpA m j q ď c 0¨R d¨v ol d´1 pBA m j q.
Since Λ 0 and Λ 1 are uniformly discrete and relatively dense, there exist constants a, b ą 0 so that for every j a¨volpA m j q ď #pΛ 0 X A m j q, #pΛ 1 X A m j q ď b¨volpA m j q.
Combining the above implies that for every j we have contradicting (2.2).

The topology on spaces of Delone sets
We consider the dynamical system pX, d, Gq, where pX, dq is a compact metric space and G is a group acting on X. The dynamical system pX, d, Gq is called minimal if every G-orbit, G.x def " tg.x | g P Gu for x P X, is dense in pX, dq. 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 0 P X is said to be uniformly recurrent if for every open neighborhood U of x 0 the set of 'return times' to U, tg P G | g.x 0 P Uu, is syndetic. As shown in [Fu,Theorem 1.15], 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 C pR d q, the space of closed subsets of pR d , ρq, 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 }¨} 8 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 topology, also known as the local rubber topology in the context of aperiodic order. It is known that D is a complete metric on C pR d q, and the space`C pR 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 is a closed subset of C pR d q and that R d acts on X by translations, the space pX, D, R d q is a compact dynamical system. We say that Λ P X is almost repetitive if for every x P R d and ε ą 0 there exists R " Rpε, xq ą 0 such that every ball Bpy, Rq in R d contains a vector v P R d that satisfies DpΛ´x, Λ´vq ă ε.
In words, for every x P R d and ε ą 0 there exists R ą 0 so that a copy of Bp0, 1{εqXpΛ´xq can be found in every R-ball, up to wiggling each point by at most ε. We also refer to [FR,Definitions 2.8,2.13,3.5] and to [LP] for distinctions between similar definitions of repetitivity. The observation in Lemma 3.1 is useful when working with the metric D in spaces of uniformly discrete point sets.
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 Delone set 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 Rpx, εq 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 pX, D, R d q is minimal. Then for every 0 ă ε ă 1 there exists R " Rpεq ą 0, so that for every Λ, Γ P X and y P R d , there exists some v P Bpy, Rq for which DpΓ, Λ´vq ă ε.
Proof. Let ε ą 0, and let Λ P X and x P R d . By minimality, the set Λ´x is uniformly recurrent. For η ą 0 denote U x η def " tΛ 1 P X | DpΛ´x, Λ 1 q ă ηu, then the set tv P R d | Λ´v P U x ε{2 u is relatively dense (syndetic). In other words, there exists R x ε{2 ą 0 such that every cube of edge-length R x ε{2 in R d contains some v P R d satisfying DpΛ´x, Λ´vq ă ε{2. By minimality again, the collection tΛ´x | x P R d u is dense in X. Thus tU x ε{2 u xPR d is an open cover of X, and by compactness there exists a finite sub-cover U x 1 ε{2 , . . . , U xn ε{2 . Then for every Γ P X these exists some j P t1, . . . , nu so that Γ P U x j ε{2 , and hence DpΓ, Λ´x j q ă ε{2. Setting R def " maxtR x 1 ε{2 , . . . , R xn ε{2 u, it follows that for every y P R d there exists some v P Bpy, R x j ε{2 q Ă Bpy, Rq such that DpΛ´x j , Λ´vq ă ε{2. Then by the triangle inequality DpΓ, Λ´vq ă ε, as required.
In Proposition 3.3 below we consider a Delone set Λ in a minimal space, and show that if sets A m in Qd grow sufficiently fast, then there exist translation vectors u m so that the patches Q m " pΛ X A m q´u m converge to a limit object that "almost" contains all of the Q m 's. The idea of the proof is simply to use the almost repetitivity property to inductively find an "almost" copy of Q m´1 inside Λ X A m , and to set u m so that it is centered accordingly, namely so that the copy we find "almost" agrees with Q m´1 . Note that every sequence of sets that grows in a reasonable sense has a subsequence that grows fast enough to satisfy conditions (1) and (2) in Proposition 3.3. Proposition 3.3. Let X be a minimal space of Delone sets in R d , Λ P X, pA m q mPN a sequence of sets in Qd and pε m q mě0 a decreasing sequence of positive constants with ε 0 ă mint1, rpΛq{2u, where rpΛq is the separation constant of Λ. For every m ě 0 choose Rpε m q satisfying Lemma 3.2 and set R m def " maxtRpε m q, 1{ε m qu. Assume that the following properties hold for every m P N: (1) There exists x m P R d such that A m Ă Bpx m , 1{2ε m q.
(2) There exists y m P R d such that Bpy m , 2R m´1 q Ă A m .
Then there exist u m P Bpy m , R m´1 q and patches Q m def " pΛXA m q´u m such that lim mÑ8 Q m " Γ P R d .Λ " X. Moreover, for every m ě 2

4)
DpQ m , Γq ă ε m´1 (3.5) and there exists c 2 ą 0 so that where c 2 depends on the dimension d and separation constant rpΛq.
Proof. First observe that by assumptions (1) and (2) holds for every m P N. In particular, the series ř 8 m"1 ε m is convergent. We define the vectors u m , and hence the patches Q m , inductively.
‚ By (1), A 1 is in particular contained in a ball of radius 1{ε 1 . Let u 1 be such that Q 1 " pΛ X A 1 q´u 1 is contained in Bp0, 1{ε 1 q. Assume that the vectors u j , and thus the patches Q j " pΛ X A j q´u j , are defined for j P t1, . . . , mu such that for every 2 ď j ď m we have (i) Bp0, R j´1 q Ă A j´uj Ă Bp0, 1{ε j q.
‚ By (2), A m`1 contains a ball of the form Bpy m`1 , 2R m q. By Lemma 3.2, let u m`1 P Bpy m`1 , R m q be a vector satisfying Thus (ii) for j " m`1 holds. Note that since Bpy m`1 , 2R m q Ă A m`1 and u m`1 P Bpy m`1 , R m q we have Bp0, R m q Ă A m`1´um`1 .
By (1), A m`1´um`1 Ă Bpx m`1´um`1 , 1{2ε m`1 q and so A m`1´um`1 contains the origin. Then by the triangle inequality, A m`1´um`1 is contained in Bp0, 1{ε m`1 q, completing the proof of (i) for j " m`1. This completes the construction of the vectors u m and the patches Q m . Next we show that the sequence pQ m q mPN is a Cauchy sequence. Fix some ε ą 0 and let M be so that 2ε M ă ε. Let m ą n ą M, and note that by property (ii) we have DpΛ´u k`1 , Λ´u k q ă ε k , for every k ě M. Then by the triangle inequality, where the third inequality follows from (3.7). By property (i), for every j P N the point sets Q j and Λ´u j in particular coincide on the ball Bp0, 1{ε j´1 q. Since m, n ą M, the sets Λ´u n and Q n coincide on Bp0, 1{εq, and similarly for Λ´u m and Q m . Therefore, relying on (3.8), for every m ą n ą M we have DpQ m , Q n q ď D´pΛ´u m q X Bp0, 1{εq, pΛ´u n q X Bp0, 1{εq¯ă ε. (3.9) Thus pQ m q mPN is a Cauchy sequence. The space pX, Dq is complete, as a compact metric space, hence the limit Γ def " lim mÑ8 Q m " lim mÑ8 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) follow immediately from the construction, see properties (i) and (ii). To see (3.5), let m P N and let k ą m be so that DpQ k , Γq ă ε m . Repeating the computations in (3.8) and (3.9) yields that DpQ m , Q k q ă 2ε m , and by (3.7) we have DpQ m , Γq ď DpQ m , Q k q`DpQ k , Γq ă 3ε m ă ε m´1 .
Finally, we prove (3.6). By (3.3) we have A m´um Ă Bp0, 1{ε m q and by (3.5) we have DpΓ, Q m`1 q ă ε m . Thus by Lemma 3.1 with A " A m´um we obtain |#pΓ X pA m´um qq´#pQ m`1 X pA m´um qq| ď c 1¨ε d m¨v ol d´1 pBA m q. (3.10) By (3.4) we have DpΛ´u m , Λ´u m`1 q ă ε m , and applying Lemma 3.1 once again we get |# ppΛ´u m`1 q X pA m´um qq´# ppΛ´u m q X pA m´um qq| ď c 1¨ε d m¨v ol d´1 pBA m q. By the definition of the Q m 's, and since A m´um Ă A m`1´um`1 by (3.3), this is exactly Combining this with (3.10) yields (3.6) and completes the proof of the theorem.

Finding patches with large discrepancy
The goal of this section is to prove the following proposition, which will be used in our proof of Theorem 1.1 in §5.
Proposition 4.1. Let Λ Ă R d be a non-uniformly spread Delone set. Then there exist a sequence pA m q mPN of sets in Qd and a sequence px m q mPN of vectors in Z d so that |#pΛ X A m q´#pΛ X pA m`xm qq| vol d´1 pBA m q mÑ8 ÝÝÝÑ 8. (4.1) Let Λ Ă R d be a Delone set. We define the central lower density and the central upper density of Λ respectively by ∆˚pΛq def " lim inf tÑ8 # pBp0, tq X Λq volpBp0, tqq ∆˚pΛq def " lim sup tÑ8 # pBp0, tq X Λq volpBp0, tqq .
If the limit lim tÑ8 # pBp0, tq X Λq {volpBp0, tqq exists, it is called the central density of Λ and is denoted by ∆pΛq. We begin with the following lemma.
Lemma 4.2. Let Λ be a Delone set, γ ą 0 and A P Qd. Then for every ε ą 0 there exists K ą 0 such that for every integer k ě K: (1) if #pΛXBp0,kqq volpBp0,kqq ě γ then the ball Bp0, kq contains A`x, a translated copy of A with x P Z d , such that #pΛ X pA`xqq volpAq ě γ´ε.
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 Bp0, kq " B r´ρs \ pBBq r`ρs , where B r´ρs , pBBq r`ρs P Qd are defined by where distpX, Y q def " inft}x´y} 8 | x P X, y P Y u. Given ε ą 0 we pick K P N large enough so that for every integer k ě K we have (4.4) and let N k def " tx P Z d | A`x Ă Bp0, kqu. By way of contradiction, assume that @x P N k : # pΛ X pA`xqq ă pγ´εqvolpAq. (4.5) Notice that the number of cubes from Q d that form A is volpAq. Then by counting the points of Λ (with multiplicity) in all the sets A`x, x P N k , the points in every unit lattice cube in B r´ρs is counted exactly volpAq times. Thus (4.6) Note that #N k ď volpBp0, kqq, then dividing both sides of (4.6) by volpAq¨volpBp0, kqq yields γ´ε ą #`Λ X B r´ρsv olpBp0, kqq (4.2) ě # pΛ X Bp0, kqq volpBp0, kqq´#`Λ X pBBq r`ρsv olpBp0, kqq Lemma 4.3. Suppose that Λ is a Delone set in R d and that ∆˚pΛq ă ∆˚pΛq. Then there exist α ă β, integers a k Ñ 8 and x k P Z d such that # pΛ X Bp0, a k qq volpBp0, a k qq ď α and # pΛ X Bpx k , a k qq volpBpx k , a k qq ě β.
Proof. By the assumption on the densities, there exist sequences a k , b l Ñ 8 so that lim kÑ8 # pΛ X Bp0, a k qq volpBp0, a k qq "α and lim lÑ8 # pΛ X Bp0, b l qq volpBp0, b l qq "β, whereα ăβ. Since Λ is uniformly discrete, and since the pd´1q-volume of the boundary of a cube grows slower than the cube's volume, we may assume that the numbers a k , b k are integers. Let δ ăβ´α 3 and fix K P N such that for every k, l ě K we have # pΛ X Bp0, a k qq volpBp0, a k qq ďα`δ and # pΛ X Bp0, b l qq volpBp0, b l qq ěβ´δ. (4.7) For every k, applying Lemma 4.2 with A " Bp0, a k q, ε "β´α 3´δ ą 0, andβ´δ in the role of γ, and combining this with (4.7), we find a large enough l " l k and x k P Z d so that Bp0, b l q contains the ball Bpx k , a k q, which satisfies # pΛ X Bpx k , a k qq volpBpx k , a k qq ě pβ´δq´ε "β´β´α 3 . (4.8) Setting α def "α`β´α 3 and β def "β´β´α 3 , the assertion follows from (4.7) and (4.8).
Proof of Proposition 4.1. Let Λ Ă R d be a non-uniformly spread Delone set. In view of Lemma 4.3 we may further assume that ∆ def " ∆pΛq exists. For α ‰ ∆´1 {d the Delone sets αZ d and Λ do not have the same central density and hence there is no BD-map between them (see e.g. [FSS,Corollary 3.2]). By our assumption on Λ, there is no BDmap between Λ and ∆´1 {d Z d as well. Applying Theorem 2.3 on these two Delone sets we obtain a sequence pA m q mPN of sets in Qd that satisfiešˇ# By passing to a subsequence of pA m q mPN we may assume that (4.9) and complete the proof using (1) of Lemma 4.2. In the case that #p∆´1 {d Z d X A m q ă #pΛ X A m q for all large values of m, the proof is similar using (2) of Lemma 4.2 instead of (1). For every m P N we pick ε m such that ε m volpA m q ă vol d´1 pBA m q (4.10) and apply Lemma 4.2 with γ " ∆´ε m , A " A m and ε " ε m . Note that since ∆pΛq " ∆ exists, the condition #pΛXBp0,kqq volpBp0,kqq ě ∆´ε m is satisfied for any sufficiently large k. By (1) of Lemma 4.2, in particular, there exists a vector x m P Z d so that # pΛ X pA m`xm qq volpA m q ě ∆´2ε m .
(4.11) By (4.9) (4.12) Note that #p∆´1 {d Z d X A m q ď ∆¨volpA m q`c¨vol d´1 pBA m q, where c depends on d and ∆, and by (4.11) we also have pΛ X pA m`xm qq ě p∆´2ε m qvolpA m q.
Then #p∆´1 {d Z d X A m q´# pΛ X pA m`xm qq ď c¨vol d´1 pBA m q`2ε m volpA m q (4.10) where c 1 depends on d and ∆. Plugging this in (4.12) completes the proof.

Proof of Theorem 1.1
Given a non-uniformly spread Delone set Λ Ă R d , let A m P Qd and x m P Z d be as in Proposition 4.1. Let ε m ą 0 be so that A m is contained in a ball of radius 1{2ε m . Passing to subsequences, by Corollary 2.4 combined with (4.1) we may assume that A m contains a ball of radius 2R m´1 , where R m is as in Proposition 3.3. We thus have 3) and so pA m q mPN and pB m q mPN both satisfy Proposition 3.3.
By (4.1), there is a sequence of constants µ m Ñ 8 such that |#pΛ X A m q´#pΛ X pA m`xm qq| " µ m¨v ol d´1 pBA m q. (5.4) Since µ m Ñ 8, by passing to a further subsequence, we may assume that µ m approaches infinity at an extremely fast rate. In particular, by defining every element in the sequence with dependence on the previous one, we may assume that 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 pA m q mPN and pB m q mPN be the sequences of sets in Qd defined in Proposition 4.1 and in (5.2), with respect to Λ. For every word ω P tA, Bu N let pC m q mPN be the sequence of sets in Qd defined by where wpmq is the m'th letter in w. Then there exists a sequence pu m q mPN of vectors in R d so that Λ ω " lim mÑ8 pΛ X C m q´u m is a Delone set in X, and @m ě 2 : |#pΛ ω X pC m´um qq´#pΛ X C m q| ď c 3¨v ol d´1 pBC m q, (5.8) where c 3 is a constant that depends on d and on the separation constant rpΛq.
Proof. Given ω P tA, Bu N , consider the sequence pC m q mPN of sets in Qd defined by (5.6). By (5.1) and (5.3), conditions (1) and (2) of Proposition 3.3 are being satisfied for pC m q mPN , with pε m q mPN as described at the beginning of this section. Applying Proposition 3.3 we obtain vectors u m satisfying (5.7), for which the sequence of patches Q m def " pΛ X C m q´u m is convergent. Setting Λ ω to be the limit set, by (3.6) of Proposition 3.3 for every m ě 2 |# pΛ ω X pC m´um qq´#Q m | ď c 3¨ε d m¨v ol d´1 pBC m q, where c 3 depends on d and on rpΛq. Clearly #Q m " #pΛ X C m q, and (5.8) follows.
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 tA, Bu N be two words that differ in infinitely many places. Then the Delone sets Λ η and Λ σ defined in Lemma 5.1 are BD-non-equivalent.
Proof. Taking a subsequence if necessary, we may assume without lose of generality that η and σ are everywhere different. We use an upper index of η or σ on elements of Qd and on vectors, e.g. C η m and u σ m , to distinguish between those elements that come from the construction of Λ η and of Λ σ in Lemma 5.1.
Denote F m def " C η m´u η m . By (5.8) for w " η we obtain @m ě 2 : |#pΛ η X F m q´#pΛ X C η m q| ď c 3¨v ol d´1 pBF m q. (5.9) Observe that for every m ě 2 there exists some v m P R d so that Indeed, assume without loss of generality that ηpmq " A and σpmq " B. Combining (5.2), (5.6) and (5.7) yields that C η m " A m , C σ m " A m`xm , u η m P Bpy m , R m´1 q and u σ m P Bpy m`xm , R m´1 q, which implies (5.10). It follows that @m ě 2 : pC σ m´u σ m q△F m Ă BF p`2R m´1 q m , and hence by (2.1) @m ě 2 : |#pΛ σ X F m q´#pΛ σ X pC σ m´u σ m qq| ď c 4¨R d m´1¨v ol d´1 pBF m q, where c 4 depends on d and on rpΛq. Again by (5.8), this time with w " σ, we obtain @m ě 2 : |#pΛ σ X F m q´#pΛ X C σ m q| ď`c 3`c4¨R d m´1˘v ol d´1 pBF m q. Combining this with (5.9), the triangle inequality yields that for every m ě 2 |#pΛ η X F m q´#pΛ σ X F m q| ě |#pΛ X C η m q´#pΛ X C σ m q|´|#pΛ η X F m q´#pΛ X C η m q|´|#pΛ σ X F m q´#pΛ X C σ m q| ě |#pΛ X C η m q´#pΛ X C σ m q|´c 5¨R d m´1¨v ol d´1 pBF m q, where c 5 depends on d and rpΛq. Since C η m " A m , C σ m " A m`xm and vol d´1 pBA m q " vol d´1 pBF m q, combined with (5.4) we have |#pΛ η X F m q´#pΛ σ X F m q| ě`µ m´c5¨R d m´1˘v ol d´1 pBF m q, and together with (5.5) we thus obtain Theorem 2.3 then implies that the sets Λ η and Λ σ are BD-non-equivalent, as required.
Proof of Theorem 1.1. Let X be a minimal space of Delone sets. If there exists a uniformly spread Λ P X, then as noted in §1 every Λ P X is uniformly spread, and (1) holds. Otherwise, there exists some Λ P X that is non-uniformly spread. Consider the equivalence relation on tA, Bu N in which η " σ if η and σ differ in only finitely many places, and let Ω Ă tA, Bu N be a set of equivalence class representatives. Since every equivalence class in this relation is countable, |Ω| " 2 ℵ 0 . For every two distinct words η, σ P Ω, Lemma 5.2 implies that Λ η and Λ σ are BD-non-equivalent, therefore BDpXq ě 2 ℵ 0 . As explained in §1 the upper bound is trivial, and so the proof is complete.