On coprime percolation, the visibility graphon, and the local limit of the GCD profile
OOn coprime percolation, the visibility graphon, andthe local limit of the gcd profile
Sébastien
Martineau ∗ February 5, 2019
Abstract
Colour an element of Z d white if its coordinates are coprime andblack otherwise. What does this colouring look like when seen from a“uniformly chosen” point of Z d ? More generally, label every elementof Z d by its gcd : what do the labels look like around a “uniform”point of Z d ? We answer these questions and generalisations of them,provide results of graphon convergence, as well as a “local/graphon”convergence. One can also investigate the percolative properties of thecolouring under study. ∗ Université Paris-Sud, [email protected] a r X i v : . [ m a t h . P R ] F e b olour an element of Z d white if its coordinates are coprime and blackotherwise. What does this colouring look like? For d = 2, this questionwas investigated in [Var99]. The starting point of the present paper is thefollowing question: what does this colouring look like when seen from a“uniformly chosen” point of Z d ? An answer has already been formulatedin [PH13] but, from the perspective adopted in the current paper, our vo-cabulary, techniques and results are more satisfactory. See the figure onpage 1.A more general version of this question goes as follows: if one labels everyelement of Z d by its gcd , what do the labels look like around a “uniform”point of Z d ? We answer this question and generalisations of it, provide re-sults of graphon convergence, as well as a “local/graphon” convergence . Byusing previous work of Vardi [Var99], we can also investigate the percolativeproperties of the colouring under study. In this paper, the set N = { , , , , , . . . } is taken to contain 0. The set ofpositive integers will be denoted by N ? = { , , , , . . . } .Let d denote a positive integer. It is well-known that the probabilitythat d numbers chosen independently and uniformly in (cid:74) , N (cid:75) are globallycoprime converges to 1 /ζ ( d ) when N goes to infinity [Dir51, Ces81, Ces83,Syl83]. Recall that on [1 , ∞ ), the Euler–Riemann ζ function is definedby ∀ s ∈ [1 , ∞ ) , ζ ( s ) := X n ≥ n − s = Y p ∈P − p − s ∈ [1 , ∞ ] , where P = { , , , , , . . . } denotes the set of prime numbers. More gen-erally, one has the following result (Theorem 459 in [HW79]). Let F be a bounded subset of R d . For every r ∈ (0 , ∞ ) , set F r := { x ∈ Z d : r − x ∈ F } . Assume that | F r | r d converges to anonzero limit when r tends to infinity.Then, one has lim r →∞ |{ x ∈ F r : gcd ( x ,...,x d )=1 }|| F r | = 1 /ζ ( d ) . (A)The study of coprime vectors of Z d , i.e. of the vectors x that satisfy gcd ( x , . . . , x d ) = 1, can be performed for its own sake. It may also bemotivated by the reducibility of fractions (the probability that a random frac-tion is irreducible is 1 /ζ (2) = π ) for d = 2 or by the visibility problem forarbitrary d . If x and y denote two distinct points of Z d , one says that x is visible from y if the line segment [ x, y ] intersects Z d only at x and y . This Essentially, the same vertex-set will at the same time be endowed with some structureof sparse graph and some structure of dense graph. x − y having a gcd equal to 1. The set of visiblepoints has been studied in various ways: see e.g. [BCZ00, BH15, BMP00,CFF19, Gar15, FM16, GHKM18, HS71, PH13, Var99].A classical corollary of (A) is the following stronger statement. Recallthat the zeta distribution of parameter s > N ? giving weight n − s ζ ( s ) to each n ∈ N ? . Let d ≥ . Let F be a bounded subset of R d . For every r ∈ (0 , ∞ ) , set F r := { x ∈ Z d : r − x ∈ F } . Assume that | F r | r d converges to a nonzero limit when r tends to infinity. Let Y r denote a uniform element in F r .Then, gcd ( Y r ) converges in distribution to a zeta distributionof parameter d , as r goes to infinity. (B)In this paper, we are interested in the following two informal questions. Colour an element of Z d white if its coordinates are coprimeand black otherwise. What does this colouring look like whenseen from a “uniformly chosen” point of Z d ? (Q1) Label every element of Z d by its gcd . What do the labels looklike around a “uniform” point of Z d ? (Q2)Even though (Q2) is stronger than (Q1), it is worthwhile to study bothquestions independently. Indeed, this leads to two strategies of differentnature, and the intermediate proposition related to (Q2) is no stronger thanthat of (Q1) — see Proposition 2.3, Proposition 2.7, and Fact 2.12.Let us now make sense of these two questions. Let X be a Polish space.Denote by Ω X the space X Z d endowed with the product topology, which isalso Polish. Let c be an element of Ω X . In (Q1), we will be interested in X = { , } and c = cop := coprime . In (Q2), we will be interested in X = N and c = gcd : x gcd ( x , . . . , x d ).If F is a nonempty finite subset of Z d , one defines the probability measure µ F,c as follows — it represents c seen from a uniform point in F . For any y ∈ Z d and ω ∈ Ω X , define τ y ω ∈ Ω X by: ∀ x ∈ Z d , ( τ y ω ) x = ω x − y . Let Y be a uniformly chosen element of F . We denote by µ F,c the distribu-tion of τ − Y c .We want to describe the limit of µ F n ,c when c is a map of interest and( F n ) is a reasonable sequence of finite subsets of Z d , such as ( (cid:74) , N (cid:75) d ),3 (cid:74) − N, N (cid:75) d ), or ( { x ∈ Z d : k x k ≤ n } ). ( ) Limits are taken in the fol-lowing sense: the space of probability measures on Ω X is endowed with theweak topology, namely the smallest topology such that for every boundedcontinuous map f : Ω X → R , the map µ R Ω X f ( ω ) dµ ( ω ) is continuous. When X is discrete, µ n converges to µ if and only if for every cylindrical event A , one has µ n [ A ] −−−−→ n →∞ µ [ A ]. Recall that a cylindrical event is an event ofthe form { ω ∈ Ω : ω | F ∈ A } , where F is a finite subset of Z d , ω | F stands for therestriction of ω to F , and A is some measurable subset of X F . When X is compact,the space of probability measures on X is compact for the weak topology. Theorem 1.1.
Let d ≥ . Let F be a bounded convex subset of R d withnonempty interior. For every r ∈ (0 , ∞ ) , set F r := { x ∈ Z d : r − x ∈ F } .Then, µ F r , gcd converges to some explicit probability measure µ ∞ , gcd when r goes to infinity. The probability measure µ ∞ , gcd is defined on page 7. It does not dependon F but only on d . Theorem 1.1 answers (Q2), hence (Q1).The remaining of the paper is organised as follows. Section 2 investigates(Q1) and (Q2). This section contains the proof of Theorem 1.1, but alsocomplementary results such as Proposition 2.4. The use of relevant arith-metic/probabilistic notions makes the core of the problem very apparent:see (cid:127) on pages 6 and 10. Section 3 provides several generalisations of theseresults. In Section 2.1, we study (Q1). In Section 2.2, we investigate (Q2). Finally,in Section 2.3, we make several comments on Propositions 2.3 and 2.7, andwe explore the percolative properties of µ ∞ , cop . In this section, we answer (Q1) by proving Theorem 2.1. Even thoughTheorem 2.1 is a particular case of Theorem 1.1, intermediate propositionsof this section cannot be deduced from the content of Section 2.2. SeeFact 2.12.Let us define µ ∞ , cop , which will turn out to be the limit of µ F n , cop when( F n ) is a reasonable sequence of finite subsets of Z d . For every prime p , let W p denote a uniformly chosen coset of p Z d in Z d , i.e. one of the p d sets ofthe form x + p Z d . Do all these choices independently. The distribution of This way of proceeding is closely related to local convergence and local weak conver-gence (also called Benjamini–Schramm convergence) : see [Bab91, BS01, DL01]. p W p is denoted by µ ∞ , cop . One can see µ ∞ , cop as a probability measureon Ω { , } by identifying a set with its indicator function. Theorem 2.1.
Let d ≥ . Let F be a bounded convex subset of R d withnonempty interior. For every r ∈ (0 , ∞ ) , set F r := { x ∈ Z d : r − x ∈ F } .Then, µ F r , cop converges to µ ∞ , cop when r goes to infinity.Remark 2.2. The convergence of µ F n , cop to µ ∞ , cop for some sequences ( F n ) of balls was conjectured by Vardi and obtained by Pleasants and Huck:see Conjecture 1 in [Var99] and Theorem 1 in [PH13]. Their vocabularyand techniques are very different from those of the present paper, and ourapproach further yields Propositions 2.3, 2.4, and 2.7. See also the two lastparagraphs of Section 2.3.We will actually prove Proposition 2.3 instead of Theorem 2.1. Say thata sequence ( F n ) of finite nonempty subsets of Z d is a Følner sequence if for every y ∈ Z d , one has | F n ∆( F n + y ) | = o ( | F n | ). It suffices to checkthis condition for a family of y ’s generating Z d as a group. This condition is alsoequivalent to lim n | ∂F n || F n | = 0, where the boundary ∂F of F ⊂ Z d is the set of theelements of Z d \ F that are adjacent to an element of F for the usual (hypercubic)graph structure of Z d . Proposition 2.3.
Let d ≥ and let ( F n ) be a Følner sequence of Z d . As-sume that µ F n ( { ω : (0 , . . . , ∈ ω } ) converges to /ζ ( d ) .Then, µ F n , cop converges to µ ∞ , cop . Theorem 2.1 immediately follows from (A) and Proposition 2.3, whichit thus suffices to prove. In Section 2.3, we will see that none of the Følnercondition and the 1 /ζ ( d )-assumption can be removed from Proposition 2.3.This proposition will itself be deduced from the following stochastic domi-nation result. If µ and ν denote two probability measures on Ω { , } , we saythat µ is stochastically dominated by ν if there is a coupling ( W , W ) of( µ, ν ) such that W ⊂ W almost surely. Proposition 2.4.
Let d ≥ and let ( F n ) be a Følner sequence of Z d . As-sume that µ F n , cop converges to some probability measure µ .Then, µ is stochastically dominated by µ ∞ , cop . The goal of the remaining of this section is to prove Proposition 2.3 andProposition 2.4.
Lemma 2.5.
Let d ≥ , let N be a positive integer, and let π : Z d → ( Z /N Z ) d denote reduction modulo N . Let ( F n ) be a Følner sequence of Z d ,and let Y n denote a uniformly chosen element of F n .Then, π ( Y n ) converges in distribution to the uniform measure on ( Z /N Z ) d . roof. Partition Z d into boxes of the form Q di =1 (cid:74) N x i , N ( x i + 1) − (cid:75) . Forevery n , say that an element x of F n is n -good if the box B containing itsatisfies B ⊂ F n . Let Y n denote a uniformly chosen element of F n . Because( F n ) is a Følner sequence, the probability that Y n is n -good converges to 1as n goes to infinity. But if Y n denotes a uniformly chosen n -good elementof F n , then π ( Y n ) is precisely uniform in ( Z /N Z ) d : one may generate Y n by picking independently a uniform element Y in (cid:74) , N − (cid:75) d and a uniformelement Y n in { x ∈ N Z d : x is n -good } , and then writing Y n = Y n + Y .The result follows. ut We will need the forthcoming Lemma 2.6, which requires the followingdefinitions of pcop and µ ∞ , pcop . Let X = { , } P . The map pcop : Z d → X is given by pcop ( x ) p = x/ ∈ p Z d . Let us now define µ ∞ , pcop . For every p ∈ P , pick a uniform coset W p of p Z d among the p d possible ones. Dothis independently for every p . One defines the random element W of Ω X via W ( x ) p := x/ ∈W p . We will denote by µ ∞ , pcop the distribution of therandom variable W . Notice that the random element W of Ω { , } definedby W ( x ) := min p W ( x ) p has distribution µ ∞ , cop . Lemma 2.6.
Let d ≥ and let ( F n ) be a Følner sequence of Z d .Then, µ F n , pcop converges to µ ∞ , pcop . Lemma 2.6 can be deduced from Lemma 2.5. We will instead deduce itfrom Lemma 2.8: see Remark 2.9. Our task is now to get Proposition 2.4from Lemma 2.6, and Proposition 2.3 from Proposition 2.4. But before doingso, I would like to explain why Proposition 2.3 does not follow directly fromLemma 2.6. (cid:127)
To deduce directly Proposition 2.3 from Lemma 2.6, one wouldneed the continuity of the map f : Ω { , } P → Ω { , } defined by f ( ω ) x =min p ω ( x ) p . Indeed, cop = f ◦ pcop . But this continuity, equivalent to thatof the min map defined from { , } P to { , } , does not hold. However,the min map is upper semicontinuous , which permits the proof of Proposi-tion 2.4. Notice that there can be no way to do “as if” f was continuous,as the Følner assumption of Lemmas 2.5, 2.6 and 2.8 does not imply that µ F n , cop converges to µ ∞ , cop — see Remark 2.13. Proof of Proposition 2.4.
Let us make the assumptions of Proposition 2.4.For Y n uniformly chosen in F n , let us consider ( τ − Y n cop , τ − Y n pcop ) ∈ Ω { , } × Ω { , } P ∼ = Ω { , }×{ , } P . As { , } × { , } P is compact, up to passing to asubsequence, we may assume that the distribution of ( τ − Y n cop , τ − Y n pcop )converges to some probability measure ρ on Ω { , } × Ω { , } P . Notice thatfor every x ∈ Z d and p ∈ P , one has pcop ( x ) p = 0 = ⇒ cop ( x ) = 0 . x and p (which range over countable sets) , the set { ( ω, ω ) : ω ( x ) ≤ ω ( x ) p } is open and closed inside Ω { , } × Ω { , } P . As a result, for ρ -almost every( ω, ω ), for every x ∈ Z d , one has ω ( x ) ≤ min p ω ( x ) p . But recall that if( W , W ) denotes a random variable with distribution ρ , then W has dis-tribution µ , W has distribution µ ∞ , pcop — by Lemma 2.6 —, and thusmin p W ( x, p ) has distribution µ ∞ , cop . Proposition 2.4 follows. ut Proof of Proposition 2.3.
Let us make the assumptions of Proposition 2.3.Up to taking a subsequence of ( F n ), we may assume that µ F n , cop convergesto some µ . We want to prove that µ = µ ∞ , cop . By Proposition 2.4, there isa monotone coupling of ( µ, µ ∞ , cop ), i.e. some coupling ρ of ( µ, µ ∞ , cop ) suchthat for ρ -almost every ( ω, ω ∞ ), one has ∀ x, ω ( x ) ≤ ω ∞ ( x ). But by the1 /ζ ( d )-assumption, one has µ ( { ω : ω ( ~
0) = 1 } ) = 1 /ζ ( d ) = Y p ∈P (1 − p − d ) = µ ∞ , cop ( { ω : ω ( ~
0) = 1 } ) . As a result, for ρ -almost every ( ω, ω ∞ ), one has ω ( ~
0) = ω ∞ ( ~ x ∈ Z d instead of ~
0. Indeed, the probabilitymeasure µ ∞ , cop is translation invariant by construction, and µ is translationinvariant because ( F n ) is a Følner sequence. Thus, µ is equal to µ ∞ , cop andProposition 2.3 is proved. ut This section is devoted to proving Theorem 1.1.Let us first define µ ∞ , gcd . For every prime p , perform the followingrandom choices: • set W p := Z d , • conditionally on ( W p , . . . , W pn − ), pick a uniform coset W pn of p n Z d among those lying inside W pn − .Do this independently for every p . We set the random p -adic valuation of avertex x in Z d to be V p ( x ) := sup { n ∈ N : x ∈ W pn } ∈ (cid:74) , ∞ (cid:75) . We definethe random gcd profile to be the random map x Q p ∈P p V p ( x ) . (Thisoccurs almost surely nowhere, but one should set Q p ∈P p V p ( x ) to be 0 whenever ∀ p, V p ( x ) = ∞ .) The distribution of the random gcd profile is denoted by µ ∞ , gcd . It is a priori a probability distribution on Ω (cid:74) , ∞ (cid:75) . By the Borel–Cantelli Lemma, for every d ≥
2, it is also a probability distribution on Ω N ,and it is as such that it will be considered from now on.We will actually prove the following proposition.7 roposition 2.7. Let d ≥ and let ( F n ) be a Følner sequence of Z d . Forevery n , let Y n denote a uniformly chosen element of F n . Assume that thedistribution of gcd ( Y n ) is tight (which implies d ≥ ) .Then, µ F n , gcd converges to µ ∞ , gcd . Theorem 1.1 immediately follows from (B) and Proposition 2.7, whichit thus suffices to prove. The forthcoming Lemma 2.8 uses the notion ofprofinite numbers. Define ˆ Z as follows:ˆ Z = lim ←− n Z /n Z = x ∈ Y n ≥ Z /n Z : ∀ ( m, n ) , m | n = ⇒ π m,n ( x n ) = x m , where π m,n denotes the morphism of reduction modulo m from Z /n Z to Z /m Z . One can see Z as a dense subgroup of ˆ Z via the injectionΦ : N ( n N + n Z ) . Elements of ˆ Z are called profinite numbers . The product topology on Q n ≥ Z /n Z induces on ˆ Z and on Z ⊂ ˆ Z the so-called profinite topology .It makes of (ˆ Z , +) a metrisable compact topological group.Recall that when p is a prime, the set of p -adic integers is definedby Z p = lim ←− n Z /p n Z . The Chinese Remainder Theorem implies that ˆ Z and Q p Z p are isomorphic as topological groups (and even as rings), anisomorphism being given byˆ Z n ( n n p n ) p ∈P ∈ Y p Z p . Let prof : Z d → ˆ Z d be defined by prof ( x ) = (Φ( x ) , . . . , Φ( x d )) ∈ ˆ Z d . Inorder to define a suitable probability measure µ ∞ , prof , notice that there is aunique Haar measure on the compact group ˆ Z ∼ = Q p Z p , which correspondsto the product of Haar measures on Z p . A Haar distributed element of Z p is a random p -adic integer ( X n ) n ≥ such that for every n ≥
1, conditionallyon X n − , the element X n is uniform among the p elements of Z /p n Z thatreduce to X n − modulo p n − .Pick a Haar distributed element Y of ˆ Z d , i.e. d independent Haar dis-tributed elements of ˆ Z . We denote by µ ∞ , prof the distribution of the followingrandom element of Ω ˆ Z d : x Y + prof ( x ) . Lemma 2.8.
Let d ≥ and let ( F n ) be a Følner sequence of Z d .Then, µ F n , prof converges to µ ∞ , prof .Proof. As ˆ Z is compact, up to passing to a subsequence, we may assumethat µ F n , prof converges to some probability measure µ .8et Y n denote a uniformly chosen element of F n , and set X n := Φ( Y n ).By Lemma 2.5, for every N , the ( N !)-component of X n converges in distri-bution to a uniform element of Z /N ! Z . As a result, X n converges in distri-bution to a Haar distributed element of ˆ Z . Therefore, if s is µ -distributed,then s (0) is Haar distributed on ˆ Z .It thus suffices to prove that µ gives probability 1 to the following set: { σ ∈ Ω ˆ Z : ∀ x ∈ Z d , σ ( x ) = σ ( ~
0) + prof ( x ) } . But this is clear as it is closedand has probability 1 for every µ F n , prof . ut Remark 2.9.
As the map f : ˆ Z → { , } P defined by f ( n ) p := n p =0 iscontinuous, Lemma 2.6 follows directly from Lemma 2.8.As we are interested in the gcd of random elements of Z , it is useful torecall what the gcd of a profinite number is. While the gcd of an element of Z is a natural number (possibly 0), the gcd of a profinite number will be a supernatural number. The set of supernatural numbers is N := (cid:74) , ∞ (cid:75) P .The set (cid:74) , ∞ (cid:75) is endowed with the usual topology (a set U is open if and onlyif for n large enough, n ∈ U ⇐⇒ ∞ ∈ U ) , and N = (cid:74) , ∞ (cid:75) P is endowed withthe product of these topologies. One can see any nonnegative integer n ∈ N as a supernatural number via the following injection:Ψ : n ( p p -adic valuation of n ) . The p -adic valuation of a profinite number n is v p ( n ) := sup { k ∈ N : n p k = 0 } ∈ (cid:74) , ∞ (cid:75) . The gcd of ( n , . . . , n d ) is the following supernatural number: p min( v p ( n ) , . . . , v p ( n d )) . Let super : Z d → N be defined by x Ψ( gcd ( x )). Given a Haar dis-tributed element Y of ˆ Z d , denote by µ ∞ , super the distribution of the followingrandom element of Ω N : x gcd ( Y + prof ( x )) . Lemma 2.10.
Let d ≥ and let ( F n ) be a Følner sequence of Z d .Then, µ F n , super converges to µ ∞ , super .Proof. As the gcd -map is continuous from ˆ Z to N , this follows directlyfrom Lemma 2.8. ut Notice that the conclusion of Lemma 2.10 is quite close to the one weare looking for. Assume further that d ≥
2. Then, as P p ∈P p − d < ∞ , theBorel–Cantelli Lemma yields that if X is µ ∞ , super -distributed, then almostsurely, every x ∈ Z d satisfies X ( x ) ∈ Ψ( N ? ). Recall that Ψ( N ? ) consists in9nitely supported elements of N = (cid:74) , ∞ (cid:75) P all the values of which are finite.If n ∈ Ψ( N ? ), then n = Ψ (cid:16)Q p ∈P p n p (cid:17) and we write χ ( n ) = Q p ∈P p n p ∈ N ? .Therefore, the random variable ( χ ( X ( x ))) x ∈ Z d has distribution µ ∞ , gcd . (cid:127) If χ was continuous and defined everywhere on N , one could directlydeduce from Lemma 2.10 that µ F n , gcd converges to µ ∞ , gcd ; but the actualproperties of χ do not allow us to perform this derivation, even in an indirectmanner. Indeed, ( F n ) being Følner and d being at least 2 do not suffice toguarantee this convergence: these conditions do not even suffice to guaranteeconvergence of µ F n , cop to µ ∞ , cop . See Remark 2.13. We need the tightnessassumption and this goes through the following easy lemma.Denote by f ] ρ the pushforward by f of a given measure ρ . Lemma 2.11.
Let X and Y be Polish spaces. Let f : X → Y be a continuousinjective map that maps every Borel subset of X to a Borel subset of Y . Let ( µ n ) n ≤∞ denote a sequence of probability measures on X . For n ≤ ∞ , set ν n := f ] µ n . Assume that ν n converges to ν ∞ and that µ n converges to someprobability measure µ .Then µ and µ ∞ are equal.Proof. By continuity of f , the sequence ν n = f ] µ n converges to f ] µ . Asthis sequence also converges to f ] µ ∞ , we have f ] µ = f ] µ ∞ . By injectivityof f , we have µ ( A ) = f ] µ ( f ( A )) = f ] µ ∞ ( f ( A )) = µ ∞ ( f ( A )), where theassumptions indeed guarantee that f ( A ) is Borel. ut Notice that in Lemma 2.11, if one does not assume that µ n convergesto some probability measure, then one cannot deduce that µ n converges to µ ∞ . A counterexample goes as follows. Take X = (0 , Y = R / Z , and set f : X → Y to be reduction modulo 1. For n ∈ N ? , set µ n = δ /n , and let µ ∞ = δ . Proof of Proposition 2.7.
Let ( F n ) and ( Y n ) be as in Proposition 2.7.By hypothesis, gcd ( Y n ) is tight. Together with the assumption that ( F n )is a Følner sequence, this implies that d ≥ x ∈ Z d , gcd ( Y n + x ) is tight. Therefore, the ( N -indexed) sequence of random vari-ables ( gcd ( Y n + x )) x ∈ Z d is tight. Up to passing to a subsequence, we maythus assume that its distribution converges to some probability measure µ on Ω N .Let X := Ω N and Y := Ω N . For n ∈ N , let µ n := µ F n , gcd . Set µ ∞ := µ ∞ , gcd , which is a well-defined probability measure on Ω N because of theBorel–Cantelli Lemma ( P p ∈P p − d < ∞ ) . Let f : X → Y be defined by f ( σ ) = (Ψ( σ x )) x ∈ Z d . This map is injective, continuous, and maps Borelsubsets of X to Borel subsets of Y . By Lemma 2.10, f ] µ n converges to f ] µ ∞ . As µ n converges to µ , Lemma 2.11 yields that µ n converges to µ ∞ ,which is the desired result. ut .3 Remarks We insist that in none of our results, we ask for the sequence ( F n ) to bemonotone, or for S n F n to be equal to Z d . The fact that | F n | tends toinfinity is a consequence of being Følner.Even though the conclusion of Proposition 2.7 implies that of Proposi-tion 2.3, it is not the case that Proposition 2.3 can be derived from Propo-sition 2.7. Indeed, one has the following fact. Fact 2.12.
For every d ≥ , there is a Følner sequence ( F n ) of Z d suchthat the following conditions hold: • if Y n denotes a uniform element of F n , the distribution of gcd ( Y n ) isnot tight, • the proportion of coprime vectors in F n converges to /ζ ( d ) .Proof. This fact can be derived from the Chinese Remainder Theoremor from the proof of Theorem 1.1. We will proceed in the second manner.If d = 1, any Følner sequence satisfies automatically the desired proper-ties. Let us thus assume that d ≥
2, so that P p ∈P p − d < ∞ . Let ε > K ∈ N . One has µ ∞ , super ( { σ : ∀ p, σ (0) p = 0 } ) = Y p ∈P (1 − p − d ) = 1 /ζ ( d ) . By translation invariance, one has Z |{ x ∈ (cid:74) , N (cid:75) d : ∀ p, σ ( x ) p = 0 }| N d d µ ∞ , super ( σ ) = 1 /ζ ( d ) . Either by ergodicity or because of Proposition 2.4, one can pick an arbitrar-ily large N such that this proportion has a positive probability to be ε -closeto 1 /ζ ( d ). Fixing such an N , one can thus find a deterministic subset U of (cid:74) , N (cid:75) d such that: (cid:12)(cid:12)(cid:12)(cid:12) | U | N d − ζ ( d ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε and µ ∞ , super ( { σ : ∀ x ∈ U, ∀ p ∈ P , σ ( x ) p = 0 } ) > . Let s be a random variable with distribution µ ∞ , super . As d ≥
2, one canpick some P ≥ max( K, N ) such that with positive probability, the followingtwo conditions hold:1. ∀ x ∈ U, ∀ p ∈ P , s ( x ) p = 0, Later in this section, we will see that µ ∞ , super is ergodic, i.e. that any translationinvariant event has probability 0 or 1. One can then use an ergodic theorem such asTheorem 1.1 in [Lin01]. ∀ x ∈ (cid:74) , N (cid:75) d , ∀ p ∈ P , ( p ≥ P = ⇒ s ( x ) p = 0).Recall that µ ∞ , super is defined in terms of the cosets W np . Let p denote aninjective map from (cid:74) , N (cid:75) d to { p ∈ P : p ≥ P } . We will use this injection tomodify s into a new random variable s . For every x ∈ (cid:74) , N (cid:75) d \ U , resample (cid:16) W p ( x ) n (cid:17) n and condition W p ( x )1 to contain x : do this independently of s and independently for every x ∈ (cid:74) , N (cid:75) d \ U . The distribution of the ensuingrandom variable s has a density relative to µ ∞ , super . As s almost surelysatisfies the following conditions:1. ∀ x ∈ U, Q p p s ( x ) p = 1,2. ∀ x ∈ (cid:74) , N (cid:75) d \ U, Q p p s ( x ) p ≥ K ,the corresponding conditions are satisfied by s with positive probability.Denote the corresponding event by E .By Theorem 1.1, the proportion of x ∈ (cid:74) , n (cid:75) d such that τ − x super satisfies E converges to a positive number, hence is positive for n large enough. Inparticular, there is some y ∈ Z d such that τ − y super satisfies E . We saythat any such y is a ( K, N, ε )-counterexample. For every n , pick some y n that is an ( n, m n , /n )-counterexample for some m n ≥ n : the sequence F n = Q di =1 (cid:74) y ni + 1 , y ni + m n (cid:75) satisfies the desired properties. ut Remark 2.13.
Likewise, the 1 /ζ ( d )-condition cannot be removed from Propo-sition 2.3: it is well-known that the Chinese Remainder Theorem impliesthat there are arbitrarily large boxes F n = x n + (cid:74) , N (cid:75) d in Z d devoid of co-prime points. This also follows from Theorem 2.1, as µ ∞ , cop has a positiveprobability to colour black the whole box (cid:74) , N (cid:75) d — use one prime per pointof the box. Such a sequence ( F n ) is Følner and yet µ F n , cop converges δ . TheFølner condition cannot be removed from Proposition 2.3 either.All the results in this paper concerning visibility extend readily to latticesin R d , as any such lattice may be mapped to Z d by a linear automorphismof R d , which preserves visibility.Notice that µ ∞ , cop and µ ∞ , gcd are not only translation invariant but also GL d ( Z )-invariant. Even though cop and gcd are indeed GL d ( Z )-invariant,the GL d ( Z )-invariance of these measures does not follow from Theorems 1.1and 2.1. One way to understand why goes as follows. Every orbit of themultiplicative group G generated by ! contains a unique point inside A := { x ∈ Z : 0 ≤ x < | x |} ∪ ( Z × { } ). Consider the colouring of Z which is defined as a the chessboard colouring on A (say white if thesum of coordinates is odd and black otherwise) , and take its unique G -invariantextension. For F n = A ∩ (cid:74) − n, n (cid:75) , this colouring seen from a uniform12oint in F n converges to the unbiased choice of one of the two chessboardcolourings of the plane: this probability measure is not G -invariant.Finally, let us mention a few questions that can be asked only withthe formalism of local limits, i.e. with µ ∞ , cop or µ ∞ , gcd instead of sim-ply one convergence result per cylindrical event as in [PH13]. For every c ∈ { cop , gcd , prof , super } , we can ask whether the measure µ ∞ ,c is ergodic— i.e. if every translation invariant measurable subset has µ ∞ ,c -probability0 or 1. These measures are indeed ergodic. For simplicity, we expose theargument for µ ∞ , cop . The Chinese Remainder Theorem guarantees the fol-lowing fact: if p , . . . , p n denote the first n primes, and if for every i ≤ n , C i denotes a coset of p i Z d in Z d , then there is a translation mapping every C i to p i Z d . As a result, whenever we consider only finitely many primes, choosingone coset per prime yields a deterministic outcome up to translation. Oneconcludes by noting that if a translation invariant probability measure on { , } Z d ×P yields ergodic measures in projection to any { , } Z d ×{ p ,...,p n } ,then the measure under study is itself ergodic. This is easily proved by mar-tingale theory, and similar reasonings are classical in the study of profiniteactions.Considering Z d to be endowed with its usual (hypercubic) graph structure,one may also ask questions of percolation theory [Gri99, LP16]: how manyinfinite white (resp. black) connected components does the colouring µ ∞ , cop yield? By ergodicity, these numbers have to be deterministic outside someevent of probability zero. One can derive from Theorem 3.3 in [Var99] that,for d = 2 hence for any d ≥
2, there is at least one infinite white connectedcomponent almost surely. One can derive from Theorem 3.4 in [Var99]that, for d = 2, there is almost surely at most one infinite white connectedcomponent and no infinite black component. Proposition 3.1 is the analog of Proposition 2.7 for an observer picked uni-formly in an affine subspace of Z d rather than in the whole space Z d .Endow (cid:74) , ∞ (cid:75) with its usual topology. Define gcd ∞ ∈ Ω (cid:74) , ∞ (cid:75) by setting gcd ∞ ( x ) := gcd ( x ). Let ( µ n ) n ≤∞ denote a sequence of probability measureson Ω (cid:74) , ∞ (cid:75) . Let V ⊂ Z d and denote by π the projection from (cid:74) , ∞ (cid:75) Z d to (cid:74) , ∞ (cid:75) V . We say that µ n converges to µ ∞ on V if π ] µ n — the pushforwardof µ n by π — converges to π ] µ ∞ when n goes to infinity.Given Γ an infinite subgroup of Z d , one defines the probability measure µ Γ ∞ , gcd ∞ by taking the definition of µ ∞ , gcd but asking furthermore at everystep that W pn intersects Γ. This corresponds to taking a Haar distributed13lement in the closure of Γ in ˆ Z d .Finally, we say that a sequence ( F n ) of nonempty finite subsets of Γ is a Følner sequence for
Γ if for every y ∈ Γ, one has | F n ∆( F n + y ) | = o ( | F n | ). Proposition 3.1.
Let Γ denote a subgroup of Z d of rank k ≥ which ismaximal among subgroups of rank k . Let ( F n ) be a Følner sequence for Γ .Let Y n denote a uniform element of F n .Then, µ F n , gcd ∞ converges to µ Γ ∞ , gcd ∞ on Z d \ Γ . If furthermore k = 1 or gcd ( Y n ) is tight, then µ F n , gcd ∞ converges to µ Γ ∞ , gcd ∞ .Remark 3.2. The view from a “uniform point” in an affine subspace Γ + y isjust the view seen from a “uniform point” in Γ shifted by − y . One may alsonotice that if one starts with a group Γ that is not maximal given its rank,then it lies in a unique such group, which is the intersection of its (rationalor real) linear span with Z d : denote it by ˜Γ. It follows from the proof ofProposition 3.1 that the following holds. Let ( F n ) be a Følner sequence forΓ. Let Y n denote a uniform element of F n . Assume that for every y ∈ ˜Γ, gcd ( Y n + y ) is tight. Then µ F n , gcd ∞ converges to µ Γ ∞ , gcd ∞ . Actually, itsuffices to make the assumption for a system of representatives of y ∈ ˜Γ forthe equivalence relation “being equal modulo Γ”, i.e. for finitely many y ’s. Corollary 3.3.
Let Γ denote a subgroup of Z d of rank k ≥ which ismaximal among subgroups of rank k . Let F denote a bounded subset of thelinear span of Γ . For every r ∈ (0 , ∞ ) , set F r := { x ∈ Γ : r − x ∈ F } .Assume that | F r | r k converges to a nonzero limit when r tends to infinity andthat ( F n ) is a Følner sequence for Γ .Then, µ F n , gcd ∞ converges to µ Γ ∞ , gcd ∞ as n goes to infinity.Proof of Proposition 3.1. Exactly as in Section 2.2, one defines µ Γ ∞ , super and proves that µ Γ F n , super converges to µ Γ ∞ , super . Let d ( · , Γ) be as in theconclusion of Lemma 3.4. Let X := Q x ∈ Z d \ Γ ψ ( (cid:74) , d ( x, Γ) (cid:75) ). As the closedsubset X ⊂ Y := N Z d \ Γ has probability 1 for every measure µ Γ F n , super , it is thecase that µ Γ ∞ , super ( X ) = 1. Therefore, one can also consider these measuresas probability measures on X : with this point of view, denote them by µ n , with n ≤ ∞ . One has that µ n converges to µ ∞ — that is for the weaktopology on probability measures on X . One way to see this is to notice thatas X is compact, every subsequence of ( µ n ) admits a converging subsequenceand then to apply Lemma 2.11 to the restriction of the identity map from X to Y . As the map f : X → (cid:74) , ∞ (cid:75) Z d \ Γ defined by f ( σ ) x := Q p ∈P p σ ( x ) p is continuous, one has that µ F n , gcd ∞ converges to µ Γ ∞ , gcd ∞ on Z d \ Γ.Let us now assume that k = 1. As P p ∈P p = ∞ , the second Borel–Cantelli Lemma yields that µ Γ ∞ , super -almost every configuration gives thelabel ∞ to every element of Γ. If Y n denotes a uniformly chosen element of F n and y some element of Γ, as k is equal to 1, one has that gcd ( Y n + y )14onverges in probability to ∞ . Together with convergence on Z d \ Γ, thisimplies that µ F n , gcd ∞ converges to µ Γ ∞ , gcd ∞ .Instead of k = 1, now assume that gcd ( Y n ) is tight. As ( F n ) is a Følnersequence for Γ, one has that for every y ∈ Γ, the random variable gcd ( Y n + y )is tight. Together with convergence on Z d \ Γ, this implies that the sequence (cid:0) µ F n , gcd ∞ (cid:1) is tight. One then concludes as in the proof of Proposition 2.7. ut Lemma 3.4.
Let ( d, k ) satisfy ≤ k ≤ d . Let Γ be a subgroup of Z d of rank k and maximal with this property. Then, there is a norm k k on R d suchthat for every N ≥ and every x ∈ N Z d , one has d ( x, Γ) > ⇒ d ( x, Γ) ≥ N, where d ( x, Γ) := min {k x − y k : y ∈ Γ } .Proof. By a change of coordinates and by maximality of Γ, one mayassume that Γ = Z k × { d − k } , in which case the lemma is clear. Notice thatany element of GL d ( Z ) maps N Z d precisely to itself. ut Proof of Corollary 3.3.
Let Y n denote a uniform element of F n . ByProposition 3.1, it suffices to assume that k ≥ gcd ( Y n )is tight. Since gcd ’s are unchanged by GL d ( Z ), we may assume that Γ isequal to Z k × { d − k } , and tightness results from (B). ut This section is devoted to (variations of) the following question: if X , . . . , X N are sampled “uniformly” and independently in Z d , can we describe the setof ( i, j )’s such that X i is visible from X j , i.e. such that gcd ( X i − X j ) = 1?A “graphon” is the data of a standard probability space ( X , P ) to-gether with a measurable function f : X → [0 ,
1] that is symmetric, i.e. sat-isfies ∀ ( x , x ) ∈ X , f ( x , x ) = f ( x , x ). Two “graphons” ( X , P X , f )and ( Y , P Y , g ) are said to induce the same graphon if, up to throw-ing away sets of measure zero, there is a measure-preserving isomorphism π : ( X , P X ) → ( Y , P Y ) such that ∀ ( x , x ) ∈ X , f ( x , x ) = g ( π ( x ) , π ( x )).A graphon is an equivalence class of “graphons” for the relation “induc-ing the same graphon”. We say that a graphon is represented by any“graphon” that induces it. See [BCL + G n = ( V n , E n ) denote a sequence of random finite graphs such that | V n | converges in probability to infinity. It is said to converge to the (de-terministic) graphon represented by ( X , P , f ) if the following holds: for every k , if ( X n , . . . , X nk ) denotes a uniform element of V kn , then the random vari-able ( { X ni ,X nj }∈ E n ) ≤ i 1] defined by δ ( x , x ) := ∀ p, x ( p ) = x ( p ) . See Figure 1. Figure 1: Visualisation of the graphon ( X , δ ) for d = 2. Af-ter suitable identification of X with [0 , δ − ( { } ) isrepresented in black. Each square at scale n is divided into p dn × p dn smaller squares, where ( p n ) denotes the sequence ofprime numbers. By the same arguments as in Section 2.1, one can prove the followingresult. Proposition 3.5. Let d ≥ and let ( F n ) be a Følner sequence of Z d . As-sume that the probability that a uniform element of F n is coprime convergesto /ζ ( d ) . Let G n denote the random graph with vertex-set F n and an edgebetween two distinct vertices if and only if the one is visible from the other.Then, G n converges to the graphon represented by ( X , δ ) . Actually, we can even get a result combining both Proposition 2.3 andProposition 3.5. Proposition 3.6. Let d ≥ and let ( F n ) be a Følner sequence of Z d . As-sume that the probability that a uniform element of F n is coprime convergesto /ζ ( d ) . Let ( X n ) denote a sequence of independent uniform elements of X . Let M ≥ and R ≥ . For every n , let ( Y nm ) ≤ m ≤ M denote M indepen-dent uniform elements in F n . Consider the following random maps: ψ n : ( (cid:74) , M, (cid:75) × (cid:74) − R, R (cid:75) d ) −→ { , } (( m , y ) , ( m , y )) Y nm + y is visible from Y nm + y , ∞ : ( (cid:74) , M, (cid:75) × (cid:74) − R, R (cid:75) d ) −→ { , } (( m , y ) , ( m , y )) ∀ p, X m ( p )+ y = X m ( p )+ y . Then, the distribution of ψ n converges to that of ψ ∞ . This theorem can be readily adapted to the whole gcd profile (one as-sumes tightness, considers maps to (cid:74) , ∞ (cid:75) rather than to { , } , and predicts the gcd of ( Y nm + y ) − ( Y nm + y )) and the case of affine subspaces.Let us conclude with a last generalisation. One may be interested inother arithmetic conditions than coprimality: for example, saying that anumber is k -free if no p k divides it, one may colour in white the k -freepoints of Z , or the points in Z d with k -free gcd . See [BMP00, Ces85,Mir47, Mir48, PH13]. One can generalise Proposition 3.6 to such contextsas follows. Recall that Z and ˆ Z can be endowed with the profinite topology.Likewise, Z d and ˆ Z d can be endowed with the product of profinite topologies,which we also refer to as the profinite topology. Example. For every k ≥ 0, the set { x ∈ Z d : gcd ( x ) is k -free } is profinitelyclosed in Z d . For k = 1, this coincides with the set { x ∈ Z d : gcd ( x ) = 1 } .Let V be a subset of Z d , and let col V := V . Let µ ∞ , col V denote thedistribution of x Y + x ∈ V , where Y is Haar distributed in ˆ Z d and V denotes the closure of the set V in ˆ Z d . Proposition 3.7. Let d ≥ and let ( F n ) be a Følner sequence of Z d . Let V denote a subset of Z d . Assume that V is profinitely closed in Z d . Assumethat | V ∩ F n || F n | converges to µ ∞ , col V ( { ω : ~ ∈ ω } ) . Let ( X m ) denote a sequenceof independent Haar distributed elements of ˆ Z d .Let M ≥ and R ≥ . For every n , let ( Y nm ) ≤ m ≤ M denote M indepen-dent uniform elements in F n . Consider the following random maps: ψ n : ( (cid:74) , M, (cid:75) × (cid:74) − R, R (cid:75) d ) −→ { , } (( m , y ) , ( m , y )) ( Y nm + y ) − ( Y nm + y ) ∈ V ,ψ ∞ : ( (cid:74) , M, (cid:75) × (cid:74) − R, R (cid:75) d ) −→ { , } (( m , y ) , ( m , y )) X m − X m + prof ( y − y ) ∈ V . Then, the distribution of ψ n converges to that of ψ ∞ . Acknowledgements At the end of a talk given by Nathanaël Enriquez,Bálint Virág asked him about the local limit of visible points: I wouldlike to thank Bálint Virág for asking this question and Nathanaël Enriquezfor letting me know of it. I am also grateful to Nathanaël Enriquez formany enthusiastic discussions about this project. 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