Divergence of separated nets with respect to displacement equivalence
aa r X i v : . [ m a t h . M G ] F e b Divergence of separated nets withrespect to displacement equivalence
Michael Dymond, Vojtěch Kaluža
We introduce a hierachy of equivalence relations on the set of separatednets of a given Euclidean space, indexed by concave increasing functions φ : (0 , ∞ ) → (0 , ∞ ) . Two separated nets are called φ -displacement equi-valent if, roughly speaking, there is a bijection between them which, forlarge radii R , displaces points of norm at most R by something of order atmost φ ( R ) . We show that the spectrum of φ -displacement equivalence spansfrom the established notion of bounded displacement equivalence , which cor-responds to bounded φ , to the indiscrete equivalence relation, coresponding to φ ( R ) ∈ Ω( R ) , in which all separated nets are equivalent. In between the twoends of this spectrum, the notions of φ -displacement equivalence are shownto be pairwise distinct with respect to the asymptotic classes of φ ( R ) for R → ∞ . We further undertake a comparison of our notion of φ -displacementequivalence with previously studied relations on separated nets. Particularattention is given to the interaction of the notions of φ -displacement equival-ence with that of bilipschitz equivalence . In the present work, we compare the metric structures of separated nets byexamining how much mappings between them displace points. The notion ofdisplacement of a mapping is defined as follows:
Definition 1.1.
Let f : A ⊆ R d → R d . We define the displacement constant of f as disp( f ) := k f − id k ∞ . If disp( f ) < ∞ , then we say that f is a mapping of bounded displacement . Research into separated nets in Euclidean spaces has broadly centred aroundthe question of to what extent any two separated nets in a Euclidean spaceare similar, as metric spaces. To formulate this question more precisely, it is
The authors acknowledge the support of Austrian Science Fund (FWF): P 30902-N35.
X, Y ⊆ R d are said to be bounded displacementequivalent , or BD equivalent , if there exists a bijection f : X → Y for which disp( f ) < ∞ . To demonstrate how constrictive BD equivalence is, we pointout that for any separated net X ⊆ R d , X is not BD equivalent to X .Hence, even linear bijections R d → R d may transform a separated net to aBD non-equivalent separated net.For the second notion, two separated nets X, Y ⊆ R d are called bilipschitzequivalent , or BL equivalent , if there is a bilipschitz bijection f : X → Y . Thisdefines a much looser form of equivalence in comparison to BD equivalence. Infact, it is a highly non-trivial question, posed by Gromov[5] in 1993, whetherBL equivalence distinguishes at all between the separated nets of a multidi-mensional Euclidean space. Moreover, we point out that BD equivalence iseasily seen to be stronger than BL equivalence.For all Euclidean spaces of dimension at least two, Gromov’s question wasanswered negatively in 1998 by Burago and Kleiner [1] and (independently)McMullen [8]; the papers [1] and [8] verify the existence of a separated netsin R d , d ≥ , which are not BL equivalent to the integer lattice.In the recent work [2], the authors introduce the notion of ω -regularity ofa separated net. Definition 1.2.
Given separated nets
X, Y ⊆ R d and a strictly increasing,concave function ω defined on a positive open interval starting at and sat-isfying lim t → ω ( t ) = 0 , a mapping f : X → Y is called a homogeneous ω -mapping if there are constants K > and a > such that k f ( y ) − f ( x ) k ≤ KRω (cid:18) k y − x k R (cid:19) for all x, y ∈ X ∩ B (0 , R ) with k y − x k < aR . The separated net X ⊆ R d is called ω -regular with respect to the separated net Y ⊆ R d if there exists abijection f : X → Y such that both f and f − are homogeneous ω -mappings.Otherwise X is called ω -irregular with respect to Y . In the case that Y = Z d ,these terms are shortened to ω -regular and ω -irregular respectively. From now on, we will refer to functions ω with the properties given inDefinition 1.2 as moduli of continuity . The function ω ( t ) = t will be called the Lipschitz modulus of continuity and functions ω ( t ) = t β with β ∈ (0 , willbe referred to as Hölder moduli of continuity . When we prescribe a modulus The reader may wish to verify this as an exercise; alternatively we note that this fact is a special caseof Proposition 3.3 of the present work.
2f continuity ω by a formula such as ω ( t ) = t β , it should be understood thatthis formula defines ω on some interval (0 , a ) with a > . The precise valueof a and indeed the behaviour of ω ( t ) for t ≥ a is irrelevant to the notions ofDefinition 1.2.It is clear that for two moduli of continuity ω , ω satisfying ω ( t ) ∈ o ( ω ( t )) for t → , the notion of ω -regularity is formally weaker than thatof ω -regularity. Further for the Lipschitz modulus of continuity ω ( t ) = t , ω -regularity of X with respect to Y is nothing other than the BL equivalenceof X and Y . Thus, the result of Burago and Kleiner and (independently)McMullen discussed above can be formulated as follows: In every Euclideanspace R d with d ≥ there exists an ω -irregular separated net for the function ω ( t ) = t .The notion of ω -regularity of separated nets is motivated by a result ofMcMullen [8, Thm. 5.1], which stands in contrast to the existence of BLnon-equivalent nets. McMullen [8] proves that for any two separated nets X and Y in Euclidean space, X is ω -regular with respect to Y for someHölder modulus of continuity ω ( t ) = t β for some β ∈ (0 , . In the work[2], the present authors investigate ω -regularity for ω lying asymptotically inbetween the Lipschitz modulus of continuity and Hölder moduli of continuity.The paper [2] proves that there are separated nets in every R d , d ≥ , whichare ω -irregular for the modulus of continuity ω ( t ) = t (cid:18) log 1 t (cid:19) α , (1)where α = α ( d ) is a positive constant determined by the dimension d ofthe space. This is formally a stronger result than the existence of BL non-equivalent separated nets. Growth of restricted displacement contants.
Looking at the value of disp( f ) for bijections f between two separated nets X and Y gives only a very crude comparison of their metric structures. Roughlyspeaking ‘most’ pairs of separated nets X and Y are BD non-equivalent, sothat disp( f ) = ∞ for every such bijection. This motivates a more subtle formof metric comparison of separated nets in Euclidean space via displacement: Definition 1.3.
Let f : A ⊆ R d → R d . We define a function (0 , ∞ ) → [0 , ∞ ) by R disp R ( f ) := ( disp( f | A ∩ B ( ,R ) ) if A ∩ B ( , R ) = ∅ , otherwise . Although we expect generally that disp( f ) = ∞ for any bijection betweentwo separated nets, so that lim R →∞ disp R ( f ) = ∞ , it remains of interestin such cases to determine the optimal asymptotic growth of disp R ( f ) as3 → ∞ among such bijections. Indeed, this allows for a more flexible notionof displacement equivalence. Definition 1.4.
Let φ : (0 , ∞ ) → (0 , ∞ ) be an increasing, concave func-tion and X and Y be separated nets of R d . We say that X and Y are φ -displacement equivalent if there exists a bijection f : X → Y for which disp R ( f ) ∈ O ( φ ( R )) . Remark 1.5.
In Definition 1.3 and Definition 1.4 it may appear that theorigin ∈ R d has a special role: it is the reference point with respect towhich the quantity disp R ( f ) is defined. It is therefore natural to ask, whethera different choice of reference point in Definition 1.3 would give rise to adifferent notion of φ -displacement equivalence in Definition 1.4. However,this is not the case, due to the conditions on the functions φ admitted inDefinition 1.4 and the inequality disp yR ( f ) ≤ disp zR + k z − y k ( f ) , where disp wR ( f ) denotes the quantity of Definition 1.3 obtained when w ∈ R d is used as the reference point instead of ∈ R d . Remark 1.6.
The reader may also ask at this point why we restrict attentiononly to concave functions φ in Definition 1.4. This is convenient, because itensures that the different functions φ that we consider are nicely comparable.For example, for concave, increasing functions φ , φ : (0 , ∞ ) → (0 , ∞ ) itholds that either φ ( R ) ∈ o ( φ ( R )) or φ ∈ Ω( φ ( R )) . Without the concavecondition this is not true.Additionally, it is the authors’ view that admitting only concave functions φ in Definition 1.4 is not a major restriction. Recall that for every increasingfunction ψ : (0 , ∞ ) → (0 , ∞ ) with φ ∈ O ( R ) there is a concave majorant,that is, a concave increasing function φ : (0 , ∞ ) → (0 , ∞ ) such that ψ ≤ φ pointwise and ψ ( R ) / ∈ o ( φ ( R )) . It is simple to construct a piecewise affineexample of such a function φ coinciding with ψ infinitely often.Observe that the concave condition in Definition 1.4 implies that φ ( R ) ∈ O ( R ) and thus superlinear functions such as φ ( R ) = R are excluded. How-ever, excluding superlinear functions φ is not any restriction because, werethey to be admitted, then the resulting notions of φ ( R ) -displacement equi-valence for all functions φ ( R ) ∈ Ω( R ) would coincide and equal the trivialequivalence relation in which all separated nets of R d are equivalent. Thislast assertion is a consequence of Proposition 2.5 of the present work. Structure of the Paper and Main Results
To finish this introduction, we outline the structure of the paper and sum-marise the main contributions of the present work.4ection 2 and 3 present preliminary results and observations which canmostly be thought of as easy consequences of the new definition of φ -displace-ment equivalence, but are nevertheless worth highlighting in view of the au-thors. In Section 2 we verify that the notions of φ -displacement equivalencegiven by Definition 1.4 are equivalence relations: Proposition 2.6.
Let φ : (0 , ∞ ) → (0 , ∞ ) be an increasing, concave function.Then the notion of φ -displacement equivalence of separated nets in R d givenby Definition 1.4 is an equivalence relation on the set of separated nets of R d . We further show that the notion of φ -displacement equivalence for φ ( R ) ∈ Ω( R ) does not distinguish between separated nets: Proposition 2.5.
Let
X, Y be two separated nets in R d . Then there is abijection f : X → Y such that disp R ( f ) , disp R ( f − ) ∈ O ( R ) . In contrast, Section 3 deals with negative results and identifies certainbarriers to φ -displacement equivalence for φ ∈ o ( R ) .Our first main result demonstrates that the notions of φ -displacementequivalence for increasing, concave functions φ : (0 , ∞ ) → (0 , ∞ ) form afine spectrum starting from the strictest form of φ -displacement equivalence,namely BD equivalence, which corresponds to φ -equivalence for bounded φ ( R ) ∈ O (1) , to the weakest form of φ -displacement equivalence, namelythat corresponding to φ ( R ) ∈ Ω( R ) . In the spectrum between O (1) and Ω( R ) we show that the notions of φ -displacement equivalence are pairwisedistinct with respect to the asymptotic classes of functions φ ( R ) for R → ∞ .We prove namely the following statement: Theorem 4.1.
Let φ : (0 , ∞ ) → (0 , ∞ ) be an increasing, concave functionwith φ ( R ) ∈ o ( R ) and X ⊆ R d be a separated net. Then there exists aseparated net Y ⊆ R d such that every bijection f : X → Y satisfies disp R ( f ) / ∈ o ( φ ( R )) and there exists a bijection g : X → Y with disp R ( g ) , disp R ( g − ) ∈ O ( φ ( R )) . Moreover, such Y can be found so that X and Y are bilipschitzequivalent. Corollary 1.7.
Let φ , φ : (0 , ∞ ) → (0 , ∞ ) be increasing, concave functionswith φ ( R ) ∈ o ( φ ( R )) . Then φ -displacement equivalence of separated netsin R d is a strictly weaker notion than that of φ -displacement equivalence. Theorem 4.1 will be proved in Section 4; Corollary 1.7 is an immediate con-sequence of Theorem 4.1. Note that Theorem 4.1 also verifies the optimalityof Proposition 2.5.The theme of Sections 5 and 6 is the comparison of the established notionof BL equivalence with the spectrum of φ -displacement equivalence for in-creasing, concave functions φ : (0 , ∞ ) → (0 , ∞ ) . We begin, in Section 5, withthe strictest form of φ -displacement equivalence, namely BD equivalence. InSection 6 we then move onto φ -displacement equivalence for unbounded φ .5e compare the notions of BL equivalence and φ -displacement equival-ence by looking at the intersection of the BL equivalence classes with theclasses of φ -displacement equivalence. The cardinality of the set of equival-ence classes of separated nets has already attracted some research attention.Magazinov [7] shows that in every Euclidean space of dimension at leasttwo, the set of BL equivalence classes of separated nets has the cardinalityof the continuum. Since BD equivalence is stronger than BL equivalence,this also implies that there are uncountably many distinct BD classes. In[4, Theorem 1.3], Frettlöh, Smilansky and Solomon also verify the existenceof uncountably many, pairwise distinct BD equivalence classes of separatednets in R . Interestingly, the class representatives of the uncountably many,pairwise distinct BD equivalence classes given in [4] all come from the sameBL equivalence class.Independently of the aforementioned works [7] and [4], we are able toverify that every Euclidean space has uncountably many, pairwise distinct BDequivalence classes of separated nets. Further, we provide new information,namely that there are uncountably many pairwise distinct BD equivalenceclasses inside each BL equivalence class. Hence, we are able to present a newresult, which we prove in Section 5: Theorem 5.1.
For every d ∈ N , every bilipschitz equivalence class of separ-ated nets in R d decomposes as a union of uncountably many pairwise distinctbounded displacement equivalence classes. For unbounded functions φ ( R ) , the analysis of the interaction of the BLclasses and the φ -displacement equivalence classes of separated nets in R d ismore challenging. In light of Theorem 5.1, the natural problem is to charac-terise the increasing, concave functions φ ( R ) ∈ o ( R ) for which φ -displacementequivalence is stronger than BL equivalence; note that Theorem 5.1 takes careof the functions φ ( R ) ∈ O (1) . In Section 6 we resolve this matter. We verify,namely, that φ -displacement is stronger than BL equivalence if and only if φ ( R ) ∈ O (1) . In particular, this means that BD equivalence is the only formof φ -displacement equivalence for which Theorem 5.1 holds.Section 6 should also be placed in the context of ω -regularity of separatednets. Recall that BL equivalence corresponds to the notion of ω -regularityfor the modulus of continuity ω ( t ) = t . For the weaker modulus of continuity ω of (1) and any function φ ( R ) ∈ O (cid:0) Rω (cid:0) R (cid:1)(cid:1) , the authors prove in [2] thatthe φ -displacement equivalence class of the integer lattice does not containany ω -irregular separated nets. This may support the following conjecture: Conjecture 1.8.
Let d ≥ , ω be a modulus of continuity in the sense ofDefinition 1.2 and φ : (0 , ∞ ) → (0 , ∞ ) be an increasing concave function.Then the class of ω -irregular separated nets in R d has non-empty intersectionwith the φ -displacement equivalence class of the integer lattice Z d if and onlyif Rω (cid:0) R (cid:1) ∈ o ( φ ( R )) . ω of (1) is precisely the result [2, Proposition 1.3] referred to above.In Section 6 of the present work, we show that for every increasing, un-bounded, concave function φ : (0 , ∞ ) → (0 , ∞ ) , the φ -displacement class ofthe integer lattice intersects distinct BL classes; in particular it contains ω -irregular separated nets for ω ( t ) = t . A matter of interest is whether everysuch φ -displacement equivalence class intersects every BL equivalence class.This question remains open, but we are able to show that every such φ -displacement equivalence class intersects uncountably many BL equivalenceclasses: Theorem 6.1.
Let d ≥ and φ : (0 , ∞ ) → (0 , ∞ ) be an unbounded, in-creasing, concave function. Then there is an uncountable family ( X ψ ) ψ ∈ Λ ofpairwise bilipschitz non-equivalent separated nets in R d for which each X ψ is φ -displacement equivalent to Z d . We point out that Theorem 6.1 is a refinement of the lower bound from [7]and is obtained entirely independently. Moreover, put together with the factthat BD equivalence is stronger than BL equivalence, Theorem 6.1 verifiesConjecture 1.8 for the special case of the Lipschitz modulus of continuity ω ( t ) = t . At this point, we also wish to state formally the characterisationannounced in the above discussion of Section 6. This result is an immediateconsequence of Theorem 5.1 and Theorem 6.1: Theorem 1.9.
Let d ≥ and φ : (0 , ∞ ) → (0 , ∞ ) be an increasing, concavefunction. Then, φ -displacement equivalence of separated nets in R d is strongerthan bilipschitz equivalence if and only if φ is bounded. Finally, we finish this article in Section 7 with a useful application of the φ -displacement equivalence spectrum. Whilst [2] verifies the existence of sep-arated nets which are ω -irregular for ω of the form (1), it leaves one import-ant issue unresolved: namely, whether ω -regularity for ω of the form (1) isdistinct from the notion of bilipschitz equivalence (that is, ω -regularity for ω ( t ) = t ). In view of the results [8, Theorem 5.1] and [1, Theorem 1.1], it isclear that there are Hölder moduli of continuity of the form ω ( t ) = t β forsome β ∈ (0 , so that for ω ( t ) = t , the notions of ω - and ω -regularityare distinct; ω -regularity is strictly weaker than ω -regularity. However, themost that can be established on the basis of the existing literature is thatthere are at least two distinct notions of ω -regularity. In particular, [2] doesnot address the issue of whether there are any moduli of continuity ω strictlyin between the Hölder moduli of continuity and the Lipschitz modulus ofcontinuity, such as ω of the form (1), which define further distinct notions of ω -regularity. This is quite unsatisfactory because it leaves open the possib-ility that the result [2, Theorem 1.2] is in fact identical to [1, Theorem 1.1]and the corresponding result in [8], although it is formally stronger.7n the present article we verify that for the modulus of continuity ω ofthe form (1), the notion of ω -regularity is strictly weaker than BL equival-ence. This confirms that the ‘highly irregular’ separated nets given in [2,Theorem 1.2] are indeed more irregular in a meaningful way than the BLnon-equivalent separated nets of McMullen [8] and Burago and Kleiner [1,Theorem 1.1]. Theorem 7.1.
Let d ≥ , α = α ( d ) be the quantity of [2, Theorem 1.2]and ω be a modulus of continuity in the sense of Definition 1.2 such that ω ( t ) = t (cid:0) log t (cid:1) α for t ∈ (0 , a ) and some a > . Then the notion of ω -regularity of separated nets in R d is strictly weaker than that of bilipschitzequivalence. Despite this progress, we are only able to increase the number of knownpairwise distinct forms of ω -regularity by one: Corollary 7.2.
For any dimension d ≥ the notions of ω -regularity of sep-arated nets in R d , according to Definition 1.2, admit at least three distinctnotions. It therefore remains an interesting research objective to expose the hier-archy of notions of ω -regularity. The authors would conjecture that, at leastfor moduli of continuity ω lying asymptotically in between the Lipschitz mod-ulus of continuity and the modulus of continuity of (1), we get a fine hier-achy of notions of ω -regularity. More precisely, we conjecture that whenevertwo moduli of continuity ω and ω satisfy ω ∈ o ( ω ( t )) , ω ( t ) ∈ Ω( t ) and ω ( t ) ∈ O (cid:0) t log (cid:0) t (cid:1) α (cid:1) for t → , then the notion of ω -regularity of separatednets in R d is strictly weaker than that of ω -regularity. Notation
Functions and Asymptotics.
Throughout the work we use the standard asymptotic notation
O, o, Ω , Θ , with the following meaning. Let f, g be twopositive real-valued functions defined on an unbounded domain in (0 , ∞ ) . Forexample, this allows for f and g to be real sequences. Then we write f ( x ) ∈ O ( g ( x )) ⇐⇒ lim sup x →∞ f ( x ) g ( x ) < ∞ ,f ( x ) ∈ o ( g ( x )) ⇐⇒ lim sup x →∞ f ( x ) g ( x ) = 0 ,f ( x ) ∈ Ω( g ( x )) ⇐⇒ g ( x ) ∈ O ( f ( x )) ,f ( x ) ∈ Θ( g ( x )) ⇐⇒ f ( x ) ∈ O ( g ( x )) and f ( x ) ∈ Ω( g ( x )) . As asserted by the title of the work [2].
8e sometimes write equations or inequalities using the above asymptoticnotation. For example, the inequalities c n ≤ n + O ( n ) ≤ O ( n ) should beinterpreted as follows: there exist sequences a n ∈ O ( n ) and b n ∈ O ( n ) suchthat c n ≤ n + a n ≤ b n . Although the symbol ω also belongs to the standardasymptotic notation, we will avoid using it in this context. The reason for thisis that we use the letter ω to denote moduli of continuity and for the notionsof ω -regularity of Definition 1.2. Since any asymptotic statement using theasymptotic ω notation can be rephrased using the little o notation, this isnot a problem.A function f : A ⊆ R → R will be called increasing if f ( t ) ≥ f ( s ) whenever s, t ∈ A and t ≥ s . If both inequalities ≥ in this condition may be replacedby the strict inequality > , then we call f strictly increasing . The notions of decreasing and strictly decreasing are defined analogously. Metric notions.
In a metric space ( M, dist M ) , a set Z ⊆ M will be called separated if inf (cid:8) dist M ( z, z ′ ) : z, z ′ ∈ Z, z = z ′ (cid:9) > , and this infimum will be referred to as the separation constant (or just the separation ) of Z (in M ). Moreover, Z will be called δ -separated if its sep-aration constant is at least δ . We will refer to the set Z as a net of M if sup (cid:26) inf z ∈ Z dist M ( z, x ) : x ∈ M (cid:27) < ∞ , and this supremum will be called the net constant of Z in M . We will call Z a θ -net of M if its net constant is at most θ .Thus, Z will be called a separated net of (or in) M if Z is both separated anda net of M . Throughout the work, we will only be concerned with separatednets of subsets of a Euclidean space R d . For a set F ⊆ R d the separated netsof F are defined according to the above discussion, where the relevant metricspace M is given by the set F together with the metric on F induced by theEuclidean distance in R d .Given two sets S, T ⊆ R d we let dist( S, T ) := inf {k t − s k : s ∈ S, t ∈ T } . In the case that S = { s } is a singleton we just write dist( s, T ) instead of dist( { s } , T ) . We write B ( x, r ) and B ( x, r ) respectively for the open andclosed balls with centre x ∈ R d and radius r ≥ . Set related notions.
The cardinality of a set A will be denoted by | A | . For m ∈ N we let [ m ] := { , , . . . , m } . We also write R + for the set of positivereal numbers. 9 easures. The symbol L will be used to denote the Lebesgue measure onthe given Euclidean space R d . Given a measurable function ρ : Q ⊆ R d → (0 , ∞ ) we let ρ L denote the measure on Q defined by ρ L ( E ) = Z E ρ d L , E ⊆ Q. Moreover, if f : Q → R d is a mapping and µ is a measure on Q , we write f ♯ µ for the pushforward measure on f ( Q ) f ♯ µ ( G ) := µ ( f − ( G )) , G ⊆ f ( Q ) . The displacement class of two separated nets.
We also introduce somenotation to conveniently capture the φ -displacement equivalences of two sep-arated nets. Definition 2.1.
Let
X, Y ⊆ R d be separated nets. By disp R ( X, Y ) , we denotethe class of increasing, concave functions φ : (0 , ∞ ) → (0 , ∞ ) for which X and Y are φ -displacement equivalent, according to Definition 1.4. Key properties of φ -displacement equivalence. The next proposition records some sufficient conditions for deriving informa-tion on the growth of disp R ( f − ) from that of disp R ( f ) . Proposition 2.2.
Let
X, Y be two separated nets in R d , φ : (0 , ∞ ) → (0 , ∞ ) be an increasing function satisfying φ ( R ) ∈ o ( R ) and φ ( R ) ≤ C φ · φ ( R/ for some constant C φ > and all R > , (2) and let f : X → Y be an injection with disp R ( f ) ≤ φ ( R ) for every R > .Then disp R ( f − ) ∈ O ( φ ( R )) . Remark.
We point out that condition (2) is satisfied whenever φ is concaveor sub-additive ( φ ( s + t ) ≤ φ ( s ) + φ ( t ) ).Proof. The assumption implies that k f ( x ) k ≥ k x k − φ ( k x k ) for every x ∈ X and that there is R > such that for every x ∈ X with k x k ≥ R itholds that k x k − φ ( k x k ) ≥ k x k / . Hence, using the assumptions on φ , wecan deduce that k x − f ( x ) k ≤ φ ( k x k ) ≤ C φ · φ (cid:18) k x k (cid:19) ≤ C φ · φ ( k f ( x ) k ) , which proves that disp R ( f − ) ∈ O ( φ ( R )) . Corollary 2.3.
Let
X, Y be two separated nets in R d and f : X → Y be aninjection with disp R ( f ) ∈ o ( R ) . Then disp R ( f − ) ∈ o ( R ) . roof. Let φ : (0 , ∞ ) → (0 , ∞ ) be a concave majorant of the function R disp R ( f ) coinciding with disp R ( f ) infinitely often (e.g., as mentioned in Re-mark 1.6). Then φ ( R ) ∈ o ( R ) . We may now apply Proposition 2.2 to φ and f to verify the corollary.The next example shows that if the assumption φ ( R ) ∈ o ( R ) in Proposi-tion 2.2 is weakened to φ ( R ) ∈ O ( R ) , then the proposition fails. It also shows,in contrast to Corollary 2.3, that no conclusion on the asymptotic class of disp R ( f − ) may be derived from the condition disp R ( f ) ∈ O ( R ) . Example 2.4.
Let ζ : (0 , ∞ ) → (0 , ∞ ) be an increasing function. Thenthere exist separated nets X, Y ⊆ R and a bijection f : X → Y such that disp R ( f ) ∈ O ( R ) and disp R ( f − ) / ∈ O ( ζ ( R )) .Proof. Let X ′ := 2 Z and Y ′ := Z . Let ψ : N → + N be any increasingfunction and define S k := { ψ ( n ) : n ∈ N , n ≥ k } for k ∈ N . Finally, we set X := X ′ ∪ S and Y := Y ′ ∪ S . Obviously, X, Y are separated nets in R .Now we can define a bijection f : X → Y as follows: f ( x ) := ( x if x ∈ X ′ ,ψ ( n − if x = ψ ( n ) . Clearly, disp R ( f ) ∈ O ( R ) , but disp ψ ( n − (cid:0) f − (cid:1) ≥ ψ ( n ) − ψ ( n − . It remainsto restrict the choice of ψ so that ψ ( n ) − ψ ( n − ≥ nζ ( ψ ( n − for all n ≥ .To finish Section 2, we prove two results announced in the introduction;their statements are repeated here for the reader’s convenience. Proposition 2.5.
Let
X, Y be two separated nets in R d . Then there is abijection f : X → Y such that disp R ( f ) , disp R ( f − ) ∈ O ( R ) .Proof. We will assume that / ∈ X, Y ; this can be ensured by an arbitrarilysmall shift. Then we observe that the condition disp R ( f ) ∈ O ( R ) is equivalentto the condition that there is C > such that k x − f ( x ) k ≤ C k x k ∀ x ∈ X. (3)Next we observe that the claim holds for X and Y if and only if there are r , r > such that it holds for r X and r Y ; assume that g : r X → r Y isa bijection and C > satisfies (3) for g instead of f . Then f : X → Y definedas f ( x ) := r g ( r x ) is also a bijection and satisfies k f ( x ) − x k = 1 r k g ( r x ) − r x k ≤ k g ( r x ) − r x k + k r x − r x k r ≤ Cr k x k + | r − r | k x k r = (cid:18) Cr + | r − r | r (cid:19) k x k x ∈ X .Moreover, note that it is enough to prove that for every X, Y there is al-ways an injection f : X → Y satisfying disp R ( f ) , disp R ( f − ) ∈ O ( R ) insteadof a bijection —the result then follows by Rado’s version of Hall’s marriagetheorem [9] from infinite graph theory. Given two injections f X : X → Y and f Y : Y → X we can define a binary relation E ⊆ X × Y so that { x, y } ∈ E if and only if f X ( x ) = y or f Y ( y ) = x . Thus, E is the union of the graphsof f X and f − Y . By Rado’s theorem there is a bijection f : X → Y suchthat ( { x, f ( x ) } ) x ∈ X ⊆ E . The condition disp R ( h ) ∈ O ( R ) for every h ∈ (cid:8) f X , f − X , f Y , f − Y (cid:9) ensures that there is C ′ > such that whenever { x, y } ∈ E , then k x k ≤ C ′ k y k and k y k ≤ C ′ k x k ; thus, disp R ( f ) , disp R ( f − ) ≤ ( C ′ + 1) R for every R > .Now let s > stand for the separation of X and b > for the net constantof Y . We choose r > such that rb < s and for every x ∈ X we find g ( x ) ∈ rY such that k x − g ( x ) k ≤ rb . As rb < s , if g ( x ) = g ( x ′ ) , then x = x ′ for any x, x ′ ∈ X . Thus, g is injective and the three observationsabove finish the proof. Proposition 2.6.
Let φ : (0 , ∞ ) → (0 , ∞ ) be an increasing, concave function.Then the notion of φ -displacement equivalence of separated nets in R d givenby Definition 1.4 is an equivalence relation on the set of separated nets of R d .Proof. Reflexivity is obvious. The symmetry of φ -displacement equivalencefollows from Proposition 2.2 if φ ( R ) ∈ o ( R ) and from Proposition 2.5 oth-erwise. To verify the transitivity, consider separated nets X, Y, Z of R d for which X and Y are φ -displacement equivalent and Y and Z are φ -displacement equivalent. Let the bijections f : X → Y and g : Y → Z witnessthis. Then g ◦ f is a bijection X → Z and there is a constant K > such that disp R ( f ) , disp R ( g ) ≤ Kφ ( R ) for all R > . Let R > and x ∈ X ∩ B ( , R ) .Then, k g ◦ f ( x ) − x k ≤ k g ( f ( x )) − f ( x ) k + k f ( x ) − x k ≤ disp R + Kφ ( R ) ( g ) + disp R ( f ) ≤ ( K + 1) φ ( R + Kφ ( R )) ≤ K ′ φ ( R ) , for some constant K ′ > independent of R and x . The existence of K ′ satisfying the last inequality is due to the conditions on φ . The present section deals with obstructions to the existence of a bijection f : X → Y between two separated nets X, Y in R d with disp R ( f ) ∈ o ( R ) .The first lemma establishes that, in the case that Y = Z d and such a bijection f : X → Z d exists, the separated net X is forced to have quite a special prop-erty. In particular it is easy to come up with examples of X not having the12roperty described in the next lemma and thus not admitting any bijection f : X → Z d with disp R ( f ) ∈ o ( R ) . Lemma 3.1.
Let X be a separated net in R d and let f : X → Z d be a bijectionsuch that disp R ( f ) ∈ o ( R ) . For any r > let µ r ( S ) := 1 r d | rS ∩ X | , S ⊆ B ( , , stand for a normalised counting measure supported on the set r X ∩ B ( , and let ( R n ) n ∈ N ⊂ R + be a sequence converging to infinity. Then the sequence ( µ R n ) n ∈ N converges weakly to L| B ( , .Proof. We write B := B ( , . Let s, b > be the separation and the netconstants of X , respectively. We set X n := R n X ∩ B and observe that each X n is sR n -separated bR n -net of B .Next we define f n : X n → R d as f n ( x ) := R n f ( R n x ) . Then the assumption disp R ( f ) ∈ o ( R ) implies that k f n − id k ∞ = 1 R n k f ◦ R n id − R n id k ∞ n →∞ −→ . (4)In other words, k f n − id k ∞ ∈ o (1) . We also observe that f n is o ( R n ) -Lipschitz: for any x, y ∈ X n it holds that k f n ( x ) − f n ( y ) k ≤ k f n ( x ) − x k + k f n ( y ) − y k + k x − y k ≤ k f n − id k ∞ + k x − y k . Applying (4), we get that k f n ( x ) − f n ( y ) k k x − y k ≤ o (1) k x − y k . As X n is sR n -separated, the right-hand side above belongs to o ( R n ) .Therefore, using Kirszbraun’s Theorem [6], each f n can be extended toan o ( R n ) -Lipschitz mapping f n : B → R d . Now for any x ∈ B we choose x n ∈ X n such that k x − x n k ≤ bR n . Considering that f n ( x n ) = f n ( x n ) and(4) we get that (cid:13)(cid:13) f n ( x ) − x (cid:13)(cid:13) ≤ (cid:13)(cid:13) f n ( x ) − f n ( x n ) (cid:13)(cid:13) + k f n ( x n ) − x n k + k x n − x k ≤ o (1) + 2 bR n n →∞ −→ , where the o (1) expression above is independent of x . This shows that f n converges uniformly to id | B .As a shortcut, we write µ n := µ R n . By an application of Prokhorov’s the-orem, we observe that the sequence ( µ n ) coverges weakly to the Lebesuge13easure on B if and only if all of its weakly convergent subsequences do.Therefore, it is enough to verify the assertion of the lemma for an arbitraryweakly convergent subsequence of ( µ n ) . We may assume, without loss of gen-erality, that this given weakly convergent subsequene is the original sequence ( µ n ) . Now we want to show that (cid:0) f n (cid:1) ♯ ( µ n ) converges weakly to L| B ; then[3, Lem. 5.6] finishes the proof. We define ε ( R ) := sup R ′ ≥ R disp R ′ ( f ) R ′ . The definition implies that ε is decreasing and from the assumption disp R ( f ) ∈ o ( R ) it follows that ε ( R n ) goes to zero as n goes to infinity. For every x ∈ X it holds that k f ( x ) k ≥ k x k − ε ( k x k ) k x k .Let M > be such that ε ( M ) < . Assume that δ ∈ (0 , , R ≥ M and x ∈ X are given such that k f ( x ) k ≤ δR . We would like to find a conditionon δ in terms of R that will ensure that k x k ≤ R . Assume that the latter isnot satisfied and consider the following inequalities: δR ≥ k f ( x ) k ≥ k x k (1 − ε ( k x k )) > R (1 − ε ( R )) . Thus, if δ ≤ (1 − ε ( R )) , we get a contradiction.The above arguments show that for every R ≥ M it holds that f (cid:0) X ∩ B ( , R ) (cid:1) ⊇ Z d ∩ B ( , (1 − ε ( R )) R ) . (5)Now we compare (cid:0) f n (cid:1) ♯ ( µ n ) to the standard normalised counting measure ν n supported on R n Z d , i.e., ν n ( S ) := 1 R dn (cid:12)(cid:12)(cid:12)(cid:12) S ∩ R n Z d (cid:12)(cid:12)(cid:12)(cid:12) for S ⊆ R d . It is clear that ν n ⇀ L . Thus, it suffices to verify that for any continuousfunction ϕ : R d → R with compact support it holds that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f n ( B ) ϕ d (cid:0) f n (cid:1) ♯ ( µ n ) − Z B ϕ d ν n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n →∞ −→ . Since (cid:0) f n (cid:1) ♯ ( µ n ) is supported on f n ( X n ) ⊂ R n Z d , we can rewrite the absolutevalue above as R dn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ f n ( X n ) ⊂ Rn Z d ϕ ( x ) − X x ∈ B ∩ Rn Z d ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . This expression can be bounded above by R dn k ϕ k ∞ (cid:12)(cid:12)(cid:12)(cid:12) f n ( X n )∆ (cid:18) B ∩ R n Z d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (6)14urther, we argue that (6) can be bounded above by k ϕ k ∞ (cid:12)(cid:12)(cid:12) R n Z d ∩ B ( , ε ( R n )) \ B ( , − ε ( R n )) (cid:12)(cid:12)(cid:12) R dn . (7)For every n ∈ N such that R n ≥ M (5) implies that f n ( X n ) ⊇ R n Z d ∩ B ( , − ε ( R n )) . Therefore, (cid:18) B ∩ R n Z d (cid:19) \ f n ( X n ) ⊆ R n Z d ∩ B \ B ( , − ε ( R n )) . (8)Using the definition of ε ( R n ) we immediately get that for any x ∈ X ∩ B ( , R n ) it holds that k f ( x ) k ≤ k x k + ε ( R n ) R n ≤ (1+ ε ( R n )) R n . Therefore, f ( X ∩ B ( , R n )) ⊆ Z d ∩ B ( , (1 + ε ( R n )) R n ) . Consequently, we deduce that f n ( X n ) ⊆ R n Z d ∩ B ( , ε ( R n )) , which together with (8) proves (7).By centering an axes-aligned cube of side length R n at each point of theset R n Z d ∩ B ( , ε ( R n )) \ B ( , − ε ( R n )) , we see that (cid:12)(cid:12)(cid:12)(cid:12) R n Z d ∩ B ( , ε ( R n )) \ B ( , − ε ( R n )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ R dn L B ∂B, ε ( R n ) + √ d R n !! . The last quantity is easily seen to be of order R dn · O (cid:16) ε ( R n ) + R n (cid:17) . Thisimplies that the upper bound of (7), and thus, also (6) go to zero as n goesto infinity.We now recall the notion of natural density of a separated net. We will seethat the natural density of separated nets is an invariant under bijections f with disp R ( f ) ∈ o ( R ) . Definition 3.2.
Let X be a separated net in R d . Then its natural density ,denoted by α ( X ) , is defined as α ( X ) := lim R →∞ (cid:12)(cid:12) X ∩ B ( , R ) (cid:12)(cid:12) L (cid:0) B ( , R ) (cid:1) , provided the limit exists; otherwise it is undefined. Proposition 3.3.
Let
X, Y be two separated nets in R d such that either α ( X ) = α ( Y ) , or exactly one of α ( X ) , α ( Y ) is not defined. Then there is nobijection f : X → Y with disp R ( f ) ∈ o ( R ) . Sometimes the term asymptotic density is used instead in the literature.
Lemma 3.4.
Let
X, Y be two separated nets in R d . Assume there is anunbounded increasing sequence ( R n ) n ∈ N ⊂ R + such that the limit L := lim n →∞ (cid:12)(cid:12) X ∩ B ( , R n ) (cid:12)(cid:12)(cid:12)(cid:12) Y ∩ B ( , R n ) (cid:12)(cid:12) is defined, but L = 1 . Then there is no bijection f : X → Y with disp R ( f ) ∈ o ( R ) .Proof. We may assume without loss of generality that
L > . Otherwisejust interchange X and Y and use Corollary 2.3. We choose C ∈ (1 , L ) and find n ∈ N such that for every n ≥ n it holds that (cid:12)(cid:12) X ∩ B ( , R n ) (cid:12)(cid:12) ≥ C (cid:12)(cid:12) Y ∩ B ( , R n ) (cid:12)(cid:12) . Because Y is a separated net, there is K > and n ∈ N such that (cid:12)(cid:12) Y ∩ B ( , KR n ) (cid:12)(cid:12) < C (cid:12)(cid:12) Y ∩ B ( , R n ) (cid:12)(cid:12) for every n ≥ n . Therefore,for every n ≥ max { n , n } we see that (cid:12)(cid:12) X ∩ B ( , R n ) (cid:12)(cid:12) > (cid:12)(cid:12) Y ∩ B ( , KR n ) (cid:12)(cid:12) ,and thus, there must be x n ∈ X ∩ B ( , R n ) such that k f ( x n ) k > KR n .Consequently, k x n − f ( x n ) k ≥ k f ( x n ) k − k x n k ≥ ( K − R n . Proof of Proposition 3.3.
The assumption on α ( X ) and α ( Y ) implies the ex-istence of a sequence ( R n ) n ∈ N satisfying the conditions of Lemma 3.4. There-fore, Proposition 3.3 follows immediately from Lemma 3.4.In view of Proposition 3.3 it is natural to ask whether for two separatednets X, Y ⊆ R d the condition that both natural densities α ( X ) and α ( Y ) arewell defined and coincide is sufficient for the existence of a bijection f : X → Y with disp R ( f ) ∈ o ( R ) . We finish this section with an example whichdemonstrates that this is not the case: Example 3.5.
There is a separated net X in R d such that α ( X ) = α ( Z d ) ,but there is no bijection f : X → Z d with disp R ( f ) ∈ o ( R ) .Proof. Fix a hyperplane H going through . We will denote the closed pos-itive and the open negative half-spaces that it determines by H + and H − ,respectively. Moreover, fix c ∈ (1 , and define X := ( c − d Z d ∩ H + ) ∪ ((2 − c ) − d Z d ∩ H − ) . Then, clearly, µ R n defined as in the statement of Lemma 3.1converges weakly to the measure c L| B ∩ H + + (2 − c ) L| B ∩ H − = L| B . On theother hand, α ( X ) = α ( Z d ) by construction. Thus, Lemma 3.1 finishes theproof. φ -displacement equivalence. In the present section we prove Theorem 4.1:16 heorem 4.1.
Let φ : (0 , ∞ ) → (0 , ∞ ) be an increasing, concave functionwith φ ( R ) ∈ o ( R ) and X ⊆ R d be a separated net. Then there exists aseparated net Y ⊆ R d such that every bijection f : X → Y satisfies disp R ( f ) / ∈ o ( φ ( R )) and there exists a bijection g : X → Y with disp R ( g ) , disp R ( g − ) ∈ O ( φ ( R )) . Moreover, such Y can be found so that X and Y are bilipschitzequivalent. Let us begin working towards a proof of Theorem 4.1. The proof will bebased on the following construction:
Construction 4.2.
Let
X, Z be separated nets in R d and let ( R i ) i ∈ N ⊂ R + be a strictly increasing sequence converging to infinity. Moreover, let φ : (0 , ∞ ) → (0 , ∞ ) be an unbounded increasing function. The aim is toconstruct a set Y in R d which will, roughly speaking, be a piecewise rescaledversion of X and such that disp R ( Y, Z ) ⊆ Ω( φ ( R )) . In the applications, wewill choose φ and ( R i ) i ∈ N in a way that will ensure that Y is a separated net.However, the construction described here is more general.Formally, we will construct Y as an image of X . For any R > weset R := min (cid:8) r ∈ R : (cid:12)(cid:12) X ∩ B ( , r ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) Z ∩ B ( , R + φ ( R )) (cid:12)(cid:12)(cid:9) . In the specialcase that Z = X we set R := R + φ ( R ) . For convenience, we also define R := R := 0 . The desired mapping g : X → R d will be radial, so wefirst define its radial part γ : [0 , ∞ ) → [0 , ∞ ) . We set γ ( R i ) := R i and inbetween these specified values the function γ interpolates linearly. Thus, γ isa piecewise linear function with breaks precisely at the points R i . Finally, wedefine g ( x ) := γ ( k x k ) k x k x and Y := g ( X ) .For later use we introduce a sequence ( c i ) i ∈ N representing the slopes of γ .That is, for every i ∈ N we require that γ ( R i ) = γ ( R i − ) + c i ( R i − R i − ) .This is equivalent to setting c i := γ ( R i ) − γ ( R i − ) R i − R i − = R i − R i − R i − R i − .We also record the maximum distance between consecutive ‘spherical lay-ers’ in X and Z . Let { ℓ < ℓ < . . . < ℓ k < . . . } := {k x k : x ∈ X } and { ℓ ′ < ℓ ′ < . . . < ℓ ′ k < . . . } := {k z k : z ∈ Z } . Additionally, we put ℓ := ℓ ′ := 0 . Then we define s X := sup { ℓ k − ℓ k − : k ∈ N } and s Z := sup (cid:8) ℓ ′ k − ℓ ′ k − : k ∈ N (cid:9) . Since X and Z are nets, s X and s Z are finite. Moreover, we set s :=max { s X , s Z } . In the rest of the present section we always refer to the notation of Con-struction 4.2 without mentioning it explicitly.
Proposition 4.3.
There exists a constant
K > depending only X, Z and φ such that the following statement holds: Assume, additionally to the assump-tions of Construction 4.2, that R > s , φ ( R ) ∈ O ( R ) and that R i ≥ KR i − for every i ∈ N . Then γ and g are bilipschitz and Y is a separated net. roof. The required conditions on
K > will be determined later in theproof. Assuming that γ is bilipschitz, it is easy to see that g is bilipschitzas well, as g is a radial map with radial part γ ; just consider the points inspherical coordinates. Moreover, a bilipschitz image in R d of a separated netin R d is a separated net in R d .The function γ is bilipschitz if and only if the sequence ( c i ) i ∈ N is boundedand bounded away from zero. By the assumption on φ there is C > suchthat R ≤ φ ( R ) + R ≤ CR for every R > s . Since
X, Z are separated nets, itholds that (cid:12)(cid:12) X ∩ B ( , R ) (cid:12)(cid:12) , (cid:12)(cid:12) Z ∩ B ( , R ) (cid:12)(cid:12) ∈ Θ( R d ) . This, in turn, implies thatthere are constants L, U > such that for every R > s we have LR ≤ R ≤ U R. (9)We note that ( R i ) i ∈ N is increasing, as both φ and ( R i ) i ∈ N are increasing. Fix i ∈ N . By (9) and the assumption on the growth of ( R i ) i ∈ N , we obtain c i = R i − R i − R i − R i − ≥ R i − R i /KR i ≥ K − KU > . Moreover, we require that
K > UL (this is the dependence of K on X, Y and φ mentioned in the statement). Then it holds that c i ≤ R i LR i − UK R i ≤ K ( LK − U ) < ∞ . Lemma 4.4.
Let φ, ( R i ) i ∈ N , X, Y, Z and s Z be as in Construction 4.2 andlet f : Y → Z be an injective mapping. Then disp R i ( f ) ≥ φ ( R i ) − s Z for every i ∈ N . Proof.
Fix i ∈ N . By Construction 4.2, it holds that (cid:12)(cid:12) Z ∩ B ( , R i + φ ( R i )) (cid:12)(cid:12) ≤ (cid:12)(cid:12) Y ∩ B ( , R i ) (cid:12)(cid:12) . This implies that disp R i ( f ) ≥ R i + φ ( R i ) − s Z − R i = φ ( R i ) − s Z . Corollary 4.5.
If, in addition to all the assumptions of Lemma 4.4, thereis
K > such that φ ( R i +1 ) ≤ Kφ ( R i ) for every i ∈ N , then disp R ( f ) ∈ Ω( φ ( R )) .Proof. Let
R > R be given and let i ∈ N be the unique index such that R i − < R ≤ R i . Then using the additional assumption and Lemma 4.4 wecan write disp R ( f ) ≥ disp R i − ( f ) ≥ φ ( R i − ) − s Z ≥ φ ( R i ) K − s Z ≥ φ ( R ) K − s Z . The last quantity is greater than, say, φ ( R ) / K for every R large enough.18 emma 4.6. Let φ, ( R i ) i ∈ N , X be as in Construction 4.2 and Y together with g : X → Y be constructed according to Construction 4.2 with Z = X . Thenfor every i ∈ N it holds that disp R i + φ ( R i ) ( g ) ≤ φ ( R i ) . Proof.
Since Z = X , we have R i = R i + φ ( R i ) for every i ∈ N accordingto Construction 4.2. This implies that c i = R i − R i − R i − R i − +( φ ( R i ) − φ ( R i − )) ≤ forevery i ∈ N . Because γ is a piecewise affine function with slopes c i ≤ and γ (0) = 0 the distance from γ to the identity is an increasing function. This,in turn, means that the displacement of g on the ball B ( , R ) is realised onthe points of X closest to the boundary of the ball. Now, we immediately getthe bound disp R i ( g ) ≤ R i − γ ( R i ) = R i + φ ( R i ) − R i = φ ( R i ) . Corollary 4.7.
If, in addition to all the assumptions of Lemma 4.6, thereis
K > such that φ ( R i +1 ) ≤ Kφ ( R i ) for every i ∈ N , then disp R ( g ) ∈ O ( φ ( R )) .Proof. Fix
R > R + φ ( R ) and choose the smallest i ∈ N such that R ≤ R i + φ ( R i ) . Then Lemma 4.6 and the additional assumption allow us toderive the bound disp R ( g ) ≤ disp R i + φ ( R i ) ( g ) ≤ φ ( R i ) ≤ Kφ ( R i − ) ≤ Kφ ( R i − + φ ( R i − )) ≤ Kφ ( R ) , where the last inequality is true thanks to the choice of i . Lemma 4.8.
For any unbounded concave function φ : (0 , ∞ ) → (0 , ∞ ) suchthat φ ( R ) ∈ o ( R ) and any given separated nets X, Z in R d there is a choiceof ( R i ) i ∈ N that satisfies all the assumptions of Proposition 4.3 and Corollar-ies 4.5 and 4.7. Namely, the minimal required properties are(i) R > s , where s is given by Construction 4.2,(ii) R i ≥ KR i − for every i ∈ N , where K > is the constant from Propos-ition 4.3,(iii) there is a constant M > such that φ ( R i +1 ) ≤ M φ ( R i ) for every i ∈ N .Proof. We fix M ≥ K large enough so that φ ( R ) < R for every R ≥ M andsuch that M > φ ( s ) . We start with the choice R := φ − ( M ) . This satisfies(i). Then we inductively define R i := φ − ( M φ ( R i − )) for every i > . Thischoice obviously satisfies (iii). Using the concavity of φ and the fact that M ≥ K > we have that φ ( KR i − ) ≤ Kφ ( R i − ) ≤ M φ ( R i − ) . Applying φ − to both sides we get (ii).19inally, we are ready to finish off the proof of Theorem 4.1: Proof of Theorem 4.1.
We may assume that φ is unbounded, otherwise wemay simply choose Y is a non-zero but small perturbation of X . Such Y is BD , and thus, also BL equivalent to X , while every bijection X → Y needsto displace the perturbed points by a non-zero distance.Lemma 4.8 provides us with a sequence ( R i ) i ∈ N for X, Z := X and φ . Weapply Construction 4.2 using these objects and obtain a set Y and a bijection g : X → Y . Proposition 4.3 says that Y is a separated net and g : X → Y witnesses the BL equivalence of X and Y . Applying Corollary 4.7 we getthat disp R ( g ) ∈ O ( φ ( R )) , from which disp R ( g − ) ∈ O ( φ ( R )) follows viaProposition 2.2. Now let f : X → Y be a bijection. By Corollary 4.5, it holdsthat disp R ( f − ) ∈ Ω( φ ( R )) and then Proposition 2.2 applies to establish disp R ( f ) / ∈ o ( φ ( R )) . The objective of the present section is to prove Theorem 5.1, whose statementwe repeat for the reader’s convenience:
Theorem 5.1.
For every d ∈ N , every bilipschitz equivalence class of separ-ated nets in R d decomposes as a union of uncountably many pairwise distinctbounded displacement equivalence classes. The proof of Theorem 5.1 is based on the following proposition:
Proposition 5.2.
Let d ∈ N , X be a separated net in R d and φ , φ : (0 , ∞ ) → (0 , ∞ ) be increasing, unbounded and concave functions such that φ i ( R ) ∈ o ( R ) for i ∈ [2] and φ ( R ) ∈ o ( φ ( R )) . Let Y , Y ⊆ R d be separated nets such that φ ∈ disp R ( X, Y ) and disp R ( X, Y ) ∩ o ( φ ( R )) = ∅ . Then Y and Y are BDnon-equivalent.Proof. Assume for a contradiction that Y and Y are BD equivalent andconsider a bijection f : Y → Y for which disp( f ) = sup x ∈ Y k f ( x ) − x k < ∞ . Let g : Y → X be a bijection for which disp R ( g ) ∈ O ( φ ) and let K > besufficiently large so that disp R ( g ) ≤ Kφ ( R ) for all R > . Then, we maydefine a bijection h : Y → X by h := g ◦ f . Let us estimate the asymptoticgrowth of disp R ( h ) : fix R > and x ∈ Y ∩ B (0 , R ) . Then f ( x ) ∈ Y ∩ B (0 , R +disp( f )) , from which it follows that k g ( f ( x )) − f ( x ) k ≤ Kφ ( R + disp( f )) .20ow we may write k h ( x ) − x k ≤ k g ( f ( x )) − f ( x ) k + k f ( x ) − x k ≤ Kφ ( R + disp( f )) + disp( f ) ≤ K ′ φ ( R ) , which is true for some K ′ > K independent of R . We deduce that h : Y → X is a bijection satisfying disp R ( h ) ∈ O ( φ ( R )) ⊆ o ( φ ( R )) , contrary to disp R ( X, Y ) ∩ o ( φ ( R )) = ∅ . Proof of Theorem 5.1.
Fix d ∈ N and a representative X of a given BL equi-valence class of separated nets in R d . Let Λ denote the set of all increasing,unbounded and concave functions (0 , ∞ ) → (0 , ∞ ) . For each φ ∈ Λ , we ap-ply Theorem 4.1 to obtain a separated net Y φ in R d belonging to the sameBL equivalence class as X and satisfying disp R ( X, Y ) ∩ o ( φ ( R )) = ∅ . Now,Proposition 5.2 verifies that the family of separated nets ( Y φ ) φ ∈ Λ containsuncountably many pairwise BD non-equivalent separated nets. In this section we prove Theorem 6.1:
Theorem 6.1.
Let d ≥ and φ : (0 , ∞ ) → (0 , ∞ ) be an unbounded, in-creasing, concave function. Then there is an uncountable family ( X ψ ) ψ ∈ Λ ofpairwise bilipschitz non-equivalent separated nets in R d for which each X ψ is φ -displacement equivalent to Z d . The proof of Theorem 6.1 will require several lemmas.
Lemma 6.2.
Let ψ , ψ : (0 , ∞ ) → (0 , ∞ ) be two increasing functions suchthat ψ ( R + K ) ∈ o ( ψ ( R )) for any fixed K ∈ N . Let ( U k ) k ∈ N be a sequenceof cubes in R d with diam U k increasing and diam U k ∈ o ( ψ ( k )) . Moreover,we assume that g , g : F k ∈ N U k → R d are mappings such that1. dist (cid:16) g ( U k ) , g (cid:16)S j = k U j (cid:17)(cid:17) ≥ ψ ( k ) for every k ∈ N ,2. dist (cid:16) g ( U k ) , g (cid:16)S j = k U j (cid:17)(cid:17) = dist( g ( U k ) , g ( U k − )) = ψ ( k ) for every k ≥ ,3. g i | U k is a translation for every i = 1 , and k ∈ N .Then for any bilipschitz mapping F : D ⊆ R d → R d there are infinitely many i ∈ N such that F D ∩ [ k ∈ N g ( U k ) ! ∩ g ( U i ) = ∅ . roof. Since F is defined only on the set D , in every application of F in thisproof the argument of F should always be intersected with D to ensure thatthe whole expression is well-defined; however, to improve the readability offormulas, we omit it.We define i ( k ) := max { i ∈ N : F ( g ( U k )) ∩ g ( U i ) = ∅} ; if the set overwhich the maximum is taken is empty, we set i ( k ) to ∞ . Let C := max (cid:8) Lip( F ) , Lip( F − ) (cid:9) . We split the proof into two cases. First, we assume that there is A ∈ N such that for every k ∈ N there is n := n ( k ) ∈ N , n ≥ k such that i ( n ) ≤ n + A . Fix k ∈ N and n = n ( k ) . Condition 2 on g implies that dist( g ( U i ( n ) ) , g ( U i ( n )+1 )) ≤ ψ ( n + A + 1) . From Condition 1 we get that dist (cid:16) F ◦ g ( U n ) , F ◦ g (cid:16)S j = n U j (cid:17)(cid:17) ≥ ψ ( n ) /C . Next, we write dist F [ j = n g ( U j ) , g ( U i ( n )+1 ) ≥ ψ ( n ) C − diam g ( U i ( n ) ) − ψ ( n + A + 1) − diam g ( U i ( n )+1 ) . Note that diam g ( U i ( n ) ) , diam g ( U i ( n )+1 ) ∈ o ( ψ ( n + A + 1)) according tothe assumptions. Thanks to the assumption ψ ( R + A + 1) ∈ o ( ψ ( R )) , weget that F (cid:16)S j ∈ N g ( U j ) (cid:17) ∩ g ( U i ( n )+1 ) = ∅ provided k (and thus n ) is largeenough. This establishes the assertion in the present case.Next we assume that for every A ∈ N it holds that i ( k ) > k + A for every k large enough. In particular, there exists k ∈ N such that i ( k ) > k forevery k ≥ k . Moreover, we assume that k is large enough so that whenever k ≥ k and F ◦ g ( U k ) ∩ g ( U i ′ ) = ∅ , we have i ′ = i ( k ) . This is possible, as F ◦ g ( U k ) ∩ g ( U i ( k ) ) = ∅ and dist (cid:16) g ( U i ( k ) ) , g (cid:16)S j = i ( k ) U j (cid:17)(cid:17) = ψ ( i ( k )) ≥ ψ ( k ) according to Condition 2 in the present case. Now it suffices to use thefact that diam F ◦ g ( U k ) ∈ o ( ψ ( k )) , which follows from the assumptions.We continue by contradiction: assume that there is i ∈ N such that forevery i ≥ i it holds that F (cid:0)S k ∈ N g ( U k ) (cid:1) ∩ g ( U i ) = ∅ . We will also assumethat i > max { i ( j ) : j ∈ N , j ≤ k , i ( j ) < ∞} . Given the property of i ( · ) proven above this means that for every i ≥ i there is k ≥ k such that i = i ( k ) . Let k ∈ N , k ≥ k be a number satisfying i ( k ) ≥ i for every k ≥ k . Next we choose K ∈ N such that either i ( k ) ≤ k + K , or i ( k ) = ∞ for every k < k . Furthermore, we choose k ∈ N , k ≥ k large enough sothat i ( k ) > k + K for every k ≥ k . In consequence, for every k ≥ k the set i ∈ N : F ◦ g [ j ≤ k U j ∩ g ( U i ) = ∅ k − k numbers within the set { k + K, . . . , k + K } . Butthis, in turn, means that there is l ∈ N , l > k such that i ( l ) ≤ k + K . At thesame time, i ( l ) > l + K ≥ k + K + 1 ; a contradiction. Lemma 6.3.
Let ρ : [0 , d → (0 , ∞ ) be a measurable function with < inf ρ ≤ sup ρ < ∞ and the property that the equation Φ ♯ ρ L = L| Φ([0 , d ) has nobilipschitz solutions Φ : [0 , d → R d . Let ( R k ) k ∈ N and ( S k ) k ∈ N be sequencesof pairwise disjoint cubes in R d such that diam R k and diam S k are unboundedand increasing and S k ⊆ R k for every k ∈ N , where S k denotes the cubewith the same midpoint as S k and sidelength twice the sidelength of S k . Foreach k ∈ N , let φ k : R d → R d denote the unique affine mapping R d → R d with scalar linear part satisfying φ k ([0 , d ) = S k . For each k ∈ N , let Υ k bea finite subset of R k such that S k ∈ N Υ k is a separated net of S k ∈ N R k andthe normalised counting measure on the set φ − k (Υ k ∩ S k ) converges weakly to ρ L . Let h : S k ∈ N Υ k → Z d be an injective mapping such that B ( h (Υ k ∩ S k ) , diam S k ) ∩ Z d ⊆ h (Υ k ) . (10) Then h is not bilipschitz. In fact, sup k ∈ N max (cid:8) Lip( h | Υ k ) , Lip(( h | Υ k ) − ) (cid:9) = ∞ . (11) Proof.
The argument of the present proof in its original form is due to Bur-ago and Kleiner; see [1, Proof of Lemma 2.1]. Moreover, a more detailedpresentation of the argument is given by the present authors in [2, Proof ofLemma 3.4]. Therefore, we present the first part of the proof here quite suc-cinctly, leaving several verifications to the reader, which may be thought ofas exercises. For further details, we refer the reader to the works [1] and [2].Observe that φ − k (2 S k ) = (cid:20) − , (cid:21) d ⊃ [0 , d = φ − k ( S k ) for all k . Suppose for a contradiction that the supremum of (11) is finite.Then, denoting by l k the sidelength of the square S k , we deduce that themappings f k := l k h ◦ φ k , extended using Kirszbraun’s theorem from Γ k := φ − k (Υ k ) ∩ (cid:20) − , (cid:21) d to the cube (cid:2) − , (cid:3) d , are uniformly Lipschitz and, after composing each f k with a translation if necessary so that the image of every f k contains , theyare also uniformly bounded. Applying the Arzelà-Ascoli theorem, we maypass to a subsequence of ( f k ) k ∈ N which converges uniformly to a Lipschitz23apping f : (cid:2) − , (cid:3) d → R d . Using the fact that each f k is bilipschitz on thefiner and finer net Γ k of (cid:2) − , (cid:3) d , we deduce that f is also bilipschitz.Let µ k denote the normalised counting measure on Γ k := φ − k (Υ k ∩ S k ) so, by hypothesis, µ k converges weakly to ρ L . We claim that the pushforwardmeasures ( f k | [0 , d ) ♯ µ k converge weakly to the Lebesgue measure on f ([0 , d ) .This claim, together with the uniform convergence of f k to f , implies that f ♯ ρ L = L| f ([0 , d ) , contrary to the hypothesis on ρ .Therefore, to complete the proof, it only remains to verify the claim, thatis, to prove that ( f k | [0 , d ) ♯ µ k converges weakly to L| f ([0 , d ) . This remainingpart of the proof is more subtle. The argument we give here is not present in[1], but is an adaptation of [2, Proof of Lemma 3.2]. Although the adaptationis quite simple, it requires good familiarity with the proof in [2] to constructit. Therefore, we provide more details here.Consider the sequence of measures ν k ( A ) := 1 l dk (cid:12)(cid:12)(cid:12)(cid:12) A ∩ l k Z d (cid:12)(cid:12)(cid:12)(cid:12) , A ⊆ R d , k ∈ N , which clearly converges weakly to the Lebesgue measure on R d . For a givencontinuous function ϕ : R d → R with compact support we need to verify (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f ([0 , d ) ϕ d ν k − Z f k ([0 , d ) ϕ d( f k | [0 , d ) ♯ µ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −→ k →∞ . (12)We bound the expression in (12) above by the sum of two terms: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f ([0 , d ) ϕ d ν k − Z f k ([0 , d ) ϕ d ν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f k ([0 , d ) ϕ d ν k − Z f k ([0 , d ) ϕ d( f k | [0 , d ) ♯ µ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (13)The first term is at most k ϕ k ∞ ν k ( f ([0 , d )∆ f k ([0 , d )) , which vanishes as k → ∞ due to the weak convergence of ν k to L , the uniform convergence of f k to f and the fact that f is bilipschitz. We do not provide further detailshere; the verification is left as an exercise with reference to [2, Lemma 3.1].The second term may be bounded above by k ϕ k ∞ l dk | A k | , where A k := f k ([0 , d ) ∩ l k Z d \ f k (Γ k ) . (14)We will argue that A k ⊆ B (cid:16) ∂f ([0 , d ) , k f k − f k ∞ (cid:17) (15)24or all k sufficiently large. Once this is established the quantity of (14) is seento be at most k ϕ k ∞ L B ∂f ([0 , d ) , k f k − f k ∞ + √ dl k !! , which converges to zero as k → ∞ . Hence, to complete the verification of theweak convergence of ( f k | [0 , d ) ♯ µ k to L| f ([0 , d ) , we prove (15).From now on we treat k as fixed but sufficiently large. Recall that thesequence of mappings f i | Γ i : Γ i → l i Z d , i ∈ N , is uniformly bilipschitz and set U := sup i ∈ N max n Lip( f i | Γ i ) , Lip( f i | Γ i − ) o < ∞ . Since the mappings f i : (cid:2) − , (cid:3) d → R d were obtained as Kirszbraun’s exten-sions of f i | Γ i , we additionally note that Lip( f i ) ≤ U for all i ∈ N . We alsowrite b for the maximum of the net constants of S ∞ i =1 Υ i ∩ S i in S ∞ i =1 S i andof S ∞ i =1 Υ i in S ∞ i =1 R i . The condition (10) translates, after application of thehomeomorphism x xl k , to B (cid:16) f k (Γ k ) , √ d (cid:17) ∩ l k Z d ⊆ f k ( φ − k (Υ k )) . At the same time, Γ k is a bl k -net of [0 , d , so that f k ([0 , d ) ⊆ B ( f k (Γ k ) , Ubl k ) .Since k is sufficiently large, it follows that A k ⊆ (cid:18) B (cid:16) f k (Γ k ) , √ d (cid:17) ∩ l k Z d (cid:19) \ f k (Γ k ) ⊆ f k ( φ − k (Υ k )) \ f k (Γ k ) . Thus, any point in A k has the form f k ( x ) for some x ∈ φ − k (Υ k \ S k ) = Γ k \ [0 , d . If f k ( x ) / ∈ f ([0 , d ) then f k ( x ) ∈ A k \ f ([0 , d ) ⊆ f k ([0 , d ) \ f ([0 , d ) , andtherefore, dist( f k ( x ) , ∂f ([0 , d )) ≤ k f k − f k ∞ .In the remaining case we have f k ( x ) ∈ f ([0 , d ) . Since f is defined at x ∈ Γ k \ [0 , d ⊆ (cid:2) − , (cid:3) d \ [0 , d and f is injective, we additionally have f ( x ) / ∈ f ([0 , d ) . Thus, we deduce that dist( f k ( x ) , ∂f ([0 , d )) ≤ k f k ( x ) − f ( x ) k ≤ k f k − f k ∞ , as required. Lemma 6.4.
Let ρ : [0 , d → (0 , ∞ ) be a measurable function with < inf ρ ≤ sup ρ < ∞ and R [0 , d ρ d L = 1 . Let ( S k ) k ∈ N be a sequence of pairwisedisjoint cubes in R d such that the sidelength l k ∈ N of S k is unbounded andincreasing. Let ( φ k ) k ∈ N denote the sequence of affine mappings φ k with scalarlinear part l k and φ k ([0 , d ) = S k . Then there exists a sequence (Ξ k ) k ∈ N offinite sets Ξ k ⊆ S k with the following properties: i) | Ξ k | = l dk for every k ∈ N ,(ii) S k ∈ N Ξ k is a separated net of S k ∈ N S k ,(iii) The sequence ( µ k ) k ∈ N , where µ k is the normalised counting measure onthe set φ − k (Ξ k ) , converges weakly to ρ L .Proof. If property (i) is omitted, the proof is contained in [1, Proof of Lemma 2.1];similar constructions are also given in [3] and [2]. Getting property (i) only re-quires taking a little extra care in the construction of [1, Proof of Lemma 2.1].Therefore, we present only minimal details here; the calculations and the veri-fication of (i)–(iii) are left to the reader.Let m k := (cid:4) √ l k (cid:5) for k ∈ N . Fix k ∈ N . We describe how to obtain the set Ξ k ⊆ S k . Consider the standard partition ( T k,i ) i ∈ [ m dk ] of the cube S k into m dk subcubes of equal size and choose a sequence ( n k,i ) i ∈ [ m dk ] satisfying n k,i ∈ ($ l dk Z φ − k ( T k,i ) ρ d L % , $ l dk Z φ − k ( T k,i ) ρ d L % + 1 ) , i ∈ [ m dk ] , X i ∈ [ m dk ] n k,i = l dk . It is now enough to define Ξ k so that | Ξ k ∩ T k,i | = n k,i for all i ∈ [ m dk ] and theseparation and net constants of Ξ k in S k may be bounded respectively belowand above independently of k . For each i ∈ [ m dk ] , we suggest the followingprescription of the set Ξ k ∩ T k,i : imagine we have a pot containing n k,i points.In the first step, we take one point out of the pot and place it at the centre ofthe cube T k,i . Assume now that j ≥ and that after j steps we have placedexactly one point from the pot at the centre of each cube in each of the thefirst j − dyadic partitions of the cube T k,i . In step j + 1 , we consider the j thdyadic partition of T k,i and arbitrarily transfer remaining points from the potonto the vacant centres of each of the dj cubes in this partition until eitherthe pot is empty or all of the dj centres are occupied. When the pot is empty,the procedure terminates and the placement of the n k,i points determines theset Ξ k ∩ T k,i . Construction 6.5.
Let ρ : [0 , d → (0 , ∞ ) be a measurable function with < inf ρ ≤ sup ρ < ∞ and R [0 , d ρ d L = 1 , l = ( l k ) k ∈ N be a strictly increas-ing sequence of natural numbers and ψ : (0 , ∞ ) → (0 , ∞ ) be an increasingfunction. We define a separated net X ( ρ, l , ψ ) as follows: Let U k := [0 , l k ] d , k ∈ N . and choose arbitrarily a mapping g ψ : F U k → R d such that g ψ and the se-quence ( U k ) k ∈ N satisfy the conditions (2) and (3) of Lemma 6.2 and ad-ditionally ∈ g ψ ( U ) . Set R k := g ψ ( U k ) for each k ∈ N . Next, fix asequence ( S k ) k ∈ N of cubes such that each S k has sidelength l k , S k ⊆ R k , ist( S k , R d \ R k ) ≥ l k and the vertices of ∂S k belong to the lattice Z d \ Z d .Let (Ξ k ) k ∈ N be the sequence of finite sets Ξ k ⊆ S k given by Lemma 6.4. Fi-nally, we define the separated net X ( ρ, l , ψ ) by X ( ρ, l , ψ ) := [ k ∈ N Ξ k ∪ Z d \ [ k ∈ N S k ! . Lemma 6.6.
Let ρ : [0 , d → (0 , ∞ ) be a measurable function with < inf ρ ≤ sup ρ < ∞ and the property that the equation Φ ♯ ρ L = L| Φ([0 , d ) has no bilipschitz solutions Φ : [0 , d → R d . Let l = ( l k ) k ∈ N be a strictlyincreasing sequence of natural numbers. Let ψ , ψ : (0 , ∞ ) → (0 , ∞ ) be in-creasing functions such that ψ ( R + K ) ∈ o ( ψ ( R )) for any fixed K ∈ N and l k ∈ o ( ψ ( k )) . Then the separated nets X i := X ( ρ, l , ψ i ) , i = 1 , , given by Construction 6.5 are bilipschitz non-equivalent.Proof. Assume that X and X are BL equivalent and let f : X → X be abijection with L := max (cid:8) Lip( f ) , Lip( f − ) (cid:9) < ∞ . Let the sequences ( U k ) k ∈ N , ( R i,k := g ψ ( U k )) k ∈ N , ( S i,k ) k ∈ N and (Ξ i,k ) k ∈ N andthe mapping g i := g ψ : F k ∈ N U k → R d be given by Construction 6.5 with thesetting ψ = ψ i for i = 1 , . In particular, this means that X i = [ k ∈ N Ξ i,k ∪ Z d \ [ k, ∈ N S i,k , i = 1 , . (16)Observe that the conditions of Lemma 6.2 are satisfied by ψ , ψ , ( U k ) k ∈ N , g , g , F := f − and D := X . Therefore, by Lemma 6.2, there is a subsequence ( U n k ) k ∈ N of ( U n ) n ∈ N such that f − (cid:0) X ∩ g (cid:0)S n ∈ N U n (cid:1)(cid:1) ∩ g ( U n k ) = ∅ forevery k ∈ N . This translates to f ( X ∩ R ,n k ) ∩ [ n ∈ N R ,n = ∅ for every k ∈ N . (17)In what follows it is occasionally necessary to assume that the first index n of the subsequence U n k is chosen sufficiently large so that, for example, aninequality like l nk > l n k holds for all k ∈ N . We will no longer mention thisexplicitly.For each k ∈ N , we set e R k := R ,n k , e S k := S ,n k and Υ k := X ∩ R ,n k .Observe that Υ k ∩ e S k = Ξ ,n k and that Υ k ∩ ( e R k \ e S k ) = Z d ∩ e R k \ e S k .Moreover, the function f | S k ∈ N Υ k has its image in Z d due to Υ k ⊆ e R k = R ,n k , (17), (16) and S ,n ⊆ R ,n . Thus, the only condition of Lemma 6.327hich is not clearly satisfied by the sequences e R k , e S k , Υ k and the function h := f | S k ∈ N Υ k : S k ∈ N Υ k → Z d is (10); we verify it shortly. However, first wepoint out that, once its conditions are verified, applying Lemma 6.3 in theabove setting gives that h = f | S k ∈ N Υ k and therefore also f is not bilipschitz,which is the desired contradiction.It therefore only remains to verify condition (10) of Lemma 6.3 for ( e R k ) k ∈ N , ( e S k ) k ∈ N , (Υ k ) k ∈ N and the function h . Let v ∈ B ( f (Υ k ∩ e S k ) , diam e S k ) ∩ Z d . We claim that v ∈ X . If v is not in X then, by the definition of X in Construction 6.5 and (16), we must have v ∈ S n ∈ N S ,n ⊂ S n ∈ N R ,n .Let b denote the net constant of X ∩ S n ∈ N R ,n in S n ∈ N R ,n and choose v ′ ∈ X ∩ S n ∈ N R ,n so that k v ′ − v k ≤ b . Let u ′ ∈ X with f ( u ′ ) = v ′ andfix a point w ∈ Υ k ∩ e S k . Then (cid:13)(cid:13) u ′ − w (cid:13)(cid:13) ≤ L (cid:13)(cid:13) f ( u ′ ) − f ( w ) (cid:13)(cid:13) ≤ L (cid:16) diam f (Υ k ∩ e S k )) + diam e S k + b (cid:17) ≤ √ dL l n k < l n k . This bound on k u ′ − w k together with w ∈ e S k and dist( e S k , R d \ e R k )) ≥ l nk implies that u ′ ∈ X ∩ e R k . But, according to (17), this in turn requires v ′ = f ( u ′ ) / ∈ (cid:0)S n ∈ N R ,n (cid:1) , contrary to the choice of v ′ . We conclude that v ∈ X .Now, we can choose z ∈ X such that v = f ( z ) . Then k z − w k ≤ L k f ( z ) − f ( w ) k ≤ diam f (Υ k ∩ e S k ) + diam e S k ≤ L √ dl n k + √ dl n k < l n k ≤ dist( e S k , R d \ e R k ) . It follows that z ∈ X ∩ e R k = Υ k and so v = f ( z ) ∈ f (Υ k ) . Lemma 6.7.
Let ρ : [0 , d → (0 , ∞ ) be a measurable function with < inf ρ ≤ sup ρ < ∞ and R [0 , d ρ d L = 1 , l = ( l k ) k ∈ N be a strictly increas-ing sequence of natural numbers and ψ : (0 , ∞ ) → (0 , ∞ ) be an increasingfunction. Let X ψ := X ( ρ, l , ψ ) be the separated net given by Construc-tion 6.5 and φ : (0 , ∞ ) → (0 , ∞ ) be an increasing, concave function suchthat φ ( ψ ( n )) ∈ Ω( l n ) . Then φ ∈ disp R ( X ψ , Z d ) .Proof. The conditions on the sidelength and the location of S k and on thesize of | Ξ k | in Construction 6.5 and Lemma 6.4(i) ensure that | X ψ ∩ S k | = (cid:12)(cid:12) Z d ∩ S k (cid:12)(cid:12) . Therefore, we may define a bijection h : X ψ → Z d as follows: onthe set X ψ \ S k ∈ N S k we define h as the identity. Finally, for each k ∈ N wedefine h | X ψ ∩ S k arbitrarily as a bijection X ψ ∩ S k → Z d ∩ S k .28he mapping h defined above clearly satisfies sup x ∈ B ( ,R ) k h ( x ) − x k = max x ∈ B ( ,R ) ∩ S k ∈ N S k k h ( x ) − x k ≤ max k : S k ∩ B ( ,R ) = ∅ diam S k . (18)On the other hand, the conditions of Construction 6.5, in particular theproperties of the mapping g ψ coming from Lemma 6.2 and ∈ g ψ ( U ) , ensurethat S k ⊆ R d \ B ( , ψ ( k )) (19)for every k > . Moreover, given R > inf x ∈ S k x k , there is a maximal n ∈ N , n ≥ such that inf x ∈ S n k x k ≤ R . We infer, using (19), that R ≥ ψ ( n ) .This, in combination with (18) implies sup x ∈ B (0 ,R ) k h ( x ) − x k φ ( R ) ≤ max k ∈ [ n ] diam S k φ ( ψ ( n )) = √ dl n φ ( ψ ( n )) ∈ O (1) . Putting together Lemmas 6.6 and 6.7 it is easy to finish the proof of The-orem 6.1:
Proof of Theorem 6.1.
Let ρ : [0 , d → (0 , ∞ ) be a measurable function with < inf ρ ≤ sup ρ < ∞ and the property that the equation Φ ♯ ρ L = L| Φ([0 , d ) has no bilipschitz solutions Φ : [0 , d → R d . Let l = ( l k ) k ∈ N be a strictlyincreasing sequence of natural numbers. Let Λ ′ denote the collection of allincreasing functions ψ : (0 , ∞ ) → (0 , ∞ ) for which φ ( ψ ( n )) ∈ Ω( l n ) and l k ∈ o ( ψ ( k )) . For each ψ ∈ Λ ′ let X ψ := X ( ρ, l , ψ ) be the separated net of R d given by Construction 6.5. Define an equivalence relation ∼ on Λ ′ by ψ ∼ ψ if X ψ and X ψ are BL equivalent. Finally, we may define Λ := Λ ′ / ∼ . Theassertions of the theorem are now readily verified using Lemmas 6.6 and6.7. ω -regularity of separated nets. Here we prove Theorem 7.1 and Corollary 7.2. The statements are repeatedfor the reader’s convenience.
Theorem 7.1.
Let d ≥ , α = α ( d ) be the quantity of [2, Theorem 1.2]and ω be a modulus of continuity in the sense of Definition 1.2 such that ω ( t ) = t (cid:0) log t (cid:1) α for t ∈ (0 , a ) and some a > . Then the notion of ω -regularity of separated nets in R d is strictly weaker than that of bilipschitzequivalence.Proof. Define φ : (0 , ∞ ) → (0 , ∞ ) by φ ( t ) = (log t ) α . Then, by Theorem 6.1there is a separated net X ⊆ R d which is BL non-equivalent to the integerlattice Z d , but for which disp R ( X, Z d ) ∩ O ( φ ( R )) = ∅ . At the same time, [2,Thm 1.2 & Prop 1.3] assert that there are ω -irregular separated nets Y ⊆ R d Y satisfy disp R ( Y, Z d ) ∩ O ( φ ( R )) = ∅ . Weconclude that the separated net X must be ω -regular. Corollary 7.2.
For any dimension d ≥ the notions of ω -regularity of sep-arated nets in R d , according to Definition 1.2, admit at least three distinctnotions.Proof. Let ω ( t ) = t (cid:0) log t (cid:1) α , where α = α ( d ) is given by Theorem 7.1, ω ( t ) = t and let X ⊆ R d be a separated net given by [2, Theorem 1.2],meaning that X is both ω - and ω -irregular. According to [8, Theorem 5.1]there exists a Hölder modulus of continuity ω ( t ) = t β for some β ∈ (0 , such that X is ω -regular. In light of Theorem 7.1, it is now clear that ω , ω and ω define pairwise distinct notions of ω -regularity. References [1] D. Burago and B. Kleiner. Separated nets in Euclidean space and Jacobi-ans of biLipschitz maps.
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