Building prescribed quantitative orbit equivalence with Z

Two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits. In particular the Ornstein-Weiss theorem implies that all inﬁnite amenable groups are orbit equivalent to the group of integers. To reﬁne this notion between inﬁnite amenable groups Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of orbit equivalence. They furthermore obtained obstructions to the existence of such equivalence using the isoperimetric proﬁle. In this article we oﬀer to answer the inverse problem (ﬁnd a group being orbit equivalent to a prescribed group with prescribed quantiﬁcation) in the case of the group of integers using the so called Følner tiling shifts introduced by Delabie et al. To do so we use the diagonal products deﬁned by Brieussel and Zheng giving groups with prescribed isoperimetric proﬁle.


introduction
Two groups are orbit equivalent if they admit free measure-preserving actions on a same standard probability space (X, µ) which share the same orbits. This notion -emerging from the seminal work of Dye [Dye59,Dye63]-can be seen as the ergodic version of the famous measure equivalence introduced by Gromov [GNR93]. A famous result of Ornstein and Weiss (see Theorem 1.2) implies that all amenable groups are orbit equivalent. In particular -unlike quasi-isometry-orbit equivalence does not preserve coarse geometric invariants.
To overcome this issue it is therefore natural to look for some refinements of this orbit equivalence notion. Assume for example that G and H are two finitely generated orbit equivalent groups over a probability space (X, µ). Recall that we can consider the Schreier graph associated to the action of G (resp. H) on X and equip it with the usual metric d S G (resp. d S H ), fixing the length of an edge to one. A first way to refine the measure equivalence is to quantify how close the two actions are by studying for all g ∈ G and h ∈ H the integrability of the two following maps When these two maps are L p we say that the groups are L p -orbit equivalent (see [BFS13] for more details). In this refined framework a famous result of Bader, Furman and Sauer [BFS13] implies that any group L 1 -orbit equivalent to a lattice in SO(n, 1) for some n ≥ 2 is virtually a lattice in SO(n, 1). This refinement also lead Bowen to prove in the appendix of [Aus16] that volume growth was invariant under L 1 -orbit equivalence. Delabie, Koivisto, Le Maître and Tessera offered in [DKLMT22] to extend this quantification to a family of functions larger than {x → x p , p ∈ [0, +∞]} (see Definition 1.3). They furthermore showed the monotonicity of the isoperimetric profile under this quantified measure equivalence definition (see Theorem 1.5). In [BZ21] Brieussel and Zheng managed to construct amenable groups with prescribed isoperimetric profile called diagonal product. Considering the monotonicity of the isoperimetric profile, the striking result of Brieussel and Zheng thus triggers a new question: instead of trying to quantify the equivalence relation between two given groups, can one find a group that is orbit equivalent to a prescribed group with a prescribed quantification?
This is the problem we address in this article. Using Brieussel-Zheng's construction we exhibit a group that is orbit equivalent to Z with a prescribed quantification (see Theorem 1.7). Comparing the obtained coupling to the constraints given by Theorem 1.5 we show that our coupling is close to being optimal for a sense of "optimal" that we make precise in Section 1.2.

Quantitative orbit equivalence
Let us recall some material from [DKLMT22]. A measure-preserving action of a discrete countable group G on a measured space (X, µ) is an action of G on X such that the map (g, x) → g · x is a Borel map and µ(E) = µ(g · E) for all E ⊆ B(X) and all g ∈ G. We will say that a measure-preserving action of G on (X, µ) is free if for almost every x ∈ X we have g · x = x if and only if g = e G .
We recall below the definition of orbit equivalence and the quantified version as introduced by Delabie, Koivisto, Le Maître and Tessera [DKLMT22]. We conclude this section by studying the relation between isoperimetric profile and orbit equivalence. All infinite amenable groups are orbit equivalent to Z.
To refine this equivalence relation and "distinguish" amenable groups we introduce the quantified version of orbit equivalence.
Recall that if a finitely generated group G acts on a space X and if S G is a finite generating set of G, we can define the Schreier graph associated to this action as being the graph whose set of vertices is X and set of edges is {(x, s · x) | s ∈ S K }. This graph is endowed with a natural metric d S G fixing the length of an edge to one. Remark that if S G is another generating set of G then there exists C > 0 such that for all x ∈ X and g ∈ G .

Definition 1.3 ([DKLMT22, Def. 2.18])
We say that an orbit equivalence coupling (X, µ) from G to H is (ϕ, ψ)-integrable if for all g ∈ G (resp. h ∈ H) there exists c g > 0 (resp. c h > 0) such that We introduce the constants c g and c h in the definition for the integrability to be independent of the choice of generating sets S G and S H . If ϕ(x) = x p we will sometimes talk of (L p , ψ)-integrability instead of (ϕ, ψ)-integrability. In particular L 0 means that no integrability assumption is made. Finally, note that every (L ∞ , ψ)-integrable coupling is (ϕ, ψ)-integrable for any increasing map ϕ : R + → R + . When ϕ = ψ we will say that the coupling is ϕ-integrable instead of (ϕ, ϕ)-integrable.
Examples 1.4 ([DKLMT22]). 1. There exists an orbit equivalence coupling between Z 4 and the Heisenberg group Heis(Z) that is L p -integrable for all p < 1. 2. Let k ∈ N * . Their exists an (L ∞ , exp)-integrable orbit equivalence coupling from the lamplighter group to the Baumslag-Solitar group BS(1, k).
More examples will be given in Section 3.1. Let us conclude on the quantification by a remark. We chose to refine orbit equivalence using the integrable point of view. But it is not the only possible sharpening. For example Kerr and Li [KL21] defined Shannon orbit equivalence: instead of looking at the integrability of distance maps they consider the Shannon entropy of partitions associated to the coupling.

Isoperimetric profile
As stated before, the orbit equivalence does not preserve the coarse geometric invariants. But the quantified version defined above allowed Delabie et al. [DKLMT22] to get a relation between the isoperimetric profiles of two orbit equivalent groups which we describe below.
Recall that if G is generated by a finite set S, the isoperimetric profile of G is defined as 1 For example the isoperimetric profile of Z verifies I Z (x) x. Remark that due to Følner criterion, a group is amenable if and only if its isoperimetric profile is unbounded. Hence we can see the isoperimetric profile as a way to measure the amenability of a group: the faster I G tends to infinity, the more amenable G is. The behaviour of the isoperimetric profile under measure equivalence coupling is given by the theorem below. If f and g are two real functions we denote f g if there exists some constant C > 0 such that f(x) = O g(Cx) as x tends to infinity. We write f g if f g and g f.
Let G and H be two finitely generated groups admitting a (ϕ, L 0 )-integrable orbit equivalence coupling. If ϕ and t/ϕ(t) are non-decreasing then This theorem provides an obstruction for finding ϕ-integrable couplings with certain functions ϕ between two amenable groups. For example for a coupling with H = Z the integrability has to verify ϕ I G . This lead the authors of [DKLMT22] to ask the following question.
Question 1.6 ([DKLMT22, Question 1.2]). Given an amenable finitely generated group G, does there exist a (I G , L 0 )-integrable orbit equivalence coupling from G to Z?
We answer the above question for a large family of maps ϕ in Theorem 1.7. We will see that the coupling we build to proof the aforementioned theorem answers Question 1.6 up to a logarithmic error.

Main results
In this paper we show the following main theorem and its corollary below.
Let us discuss the optimality of this result. Consider a (ϕ, L 0 )-integrable orbit equivalence coupling from some group G to Z. By Theorem 1.5 it verifies ϕ • I Z I G . In particular since I Z (x) x, we can not have a better integrability than ϕ(x) I G . Since I G ρ • log our above theorem is optimal up to a logarithmic error. We discuss this in more length in Section 5. main ingredients The main tools of the proof of Theorem 1.7 are Brieussel-Zheng's diagonal products (see Section 2) and Følner tiling shifts (see Section 3). We show that a diagonal product ∆ admits a coupling with Z satisfying Theorem 1.7. To prove it we use the integrability criterion given by Theorem 3.5 and involving Følner tiling shifts.
Therefore we compute in Section 3.2 a Følner tiling shift (Σ n ) n for ∆. We also estimate the tiles' diameter and the proportion of elements in the boundary. We construct a Følner tiling shift for Z in Section 4.1 and show that these two tiling shifts verify Theorem 3.5.
Let us now consider the possible generalisations of this result to other groups than the group of integers. To do so we can use the composition of couplings described in [DKLMT22, Section 2].
Given the above theorem, once we have a measure equivalence coupling from Z to a group H we can compose the two couplings to obtain a measure equivalence from G to H. If the growth of the isoperimetric profile of H is close to the one of Z, the integrability of the obtained coupling will be close to the optimal one given by Theorem 1.5. It is for example the case when H = Z d .
structure of the paper In Section 2 we present the diagonal products introduced by Brieussel and Zheng. We recall some of the properties shown in [BZ21] and compute Følner sequences. Section 3 is devoted to Følner tiling shifts. These tools built by Delabie et al. [DKLMT22] allow us to construct and quantify an orbit equivalence coupling between two groups. In this section we also construct Følner tiling shifts for diagonal products ∆. We show our main theorem in Section 4 combining the results of the two previous sections. Finally we discuss the limits of this construction and some open problems in Section 5.
I acknowledgements I would like to thank Romain Tessera and Jérémie Brieussel, under whose supervision the work presented in this article was carried out. I thank them for suggesting the topic, sharing their precious insights and for their many useful advice. I also thank the anonymous referee for their remarks and corrections.

diagonal products of lamplighter groups
We recall here necessary material from [BZ21] concerning the definition of Brieussel-Zheng's diagonal products. We give the definition of such a group, recall and prove some results concerning the range (see Definition 2.7) of an element and use it to identify a Følner sequence. Finally we present in Section 2.3 the tools needed to recover such a diagonal product starting with a prescribed isoperimetric profile.

Definition of diagonal products
Recall that the wreath product of a group G with Z denoted G Z is defined as G Z := ⊕ m∈Z G Z. An element of G Z is a pair (f, t) where f is a map from Z to G with finite support and t belongs to Z. We refer to f as the lamp configuration and t as the cursor. Finally we denote by supp(f) the support of f which is defined as supp(f) := {x ∈ Z | f(x) = e G }.

General definition
Let A and B be two finite groups. Let (Γ m ) m∈N be a sequence of finite groups such that each Γ m admits a generating set of the form A m ∪B m where A m and B m are finite subgroups of Γ m isomorphic respectively to A and B. For a ∈ A we denote a m the copy of a in A m and similarly for B m . Finally let (k m ) m∈N be a sequence of integers such that k m+1 ≥ 2k m for all m. We define ∆ m = Γ m Z and endow it with the generating set

Definition 2.1
The Brieussel-Zheng diagonal product associated to (Γ m ) m∈N and (k m ) m∈N is the subgroup ∆ of ( m Γ m ) Z generated by The group ∆ is uniquely determined by the sequences (Γ m ) m∈N and (k m ) m∈N . Let us give an illustration of what an element in such a group looks like. We will denote by g the sequence (g m ) m∈N .
Example 2.2. We represent in Figure

The expanders case
In this article we will restrict ourselves to a particular familiy of groups (Γ m ) m∈N called expanders. Recall that (Γ m ) m∈N is said to be a sequence of expanders if the sequence of diameters (diam (Γ m )) m∈N is unbounded and if there exists c 0 > 0 such that for all m ∈ N and all n ≤ |Γ m |/2 the isoperimetric profile verifies I Γm (n) ≤ c 0 .
When talking about diagonal products we will always make the following assumptions. We refer to [BZ21, Example 2.3] for an explicit example of diagonal product verifying (H). Cursor (a 0 ,b 0 ) g 0 a 1 b 1 g 1 a 2 b 2 g 2 ⋮ a n b n g n Figure 1: Representation of g, t = (a m δ 0 ) m , 0 (b m δ km ) m , 0 (0, 3) when k m = 2 m .

Hypothesis (H)
• (k m ) m and (l m ) m are sub-sequences of geometric sequences; Recall (see [BZ21,page 9]) that in this case there exist c 1 , c 2 > 0 such that, for all m Finally we adopt the convention of [BZ21, Notation 2.2] and allow (k m ) m∈N to take the value +∞. In this case ∆ s is the trivial group. In particular when k 1 = +∞ the diagonal product ∆ corresponds to the usual lamplighter (A × B) Z.

Relative commutators subgroups
Example 2.3. Let (g, 3) be the element described in Figure 1. Then the only non-trivial value of θ 0 (g 0 ) is θ 0 (g 0 (0)) = (a 0 , b 0 ). If m > 0 then the only non trivial values of θ m (g m ) are θ m (g m (0)) = (a m , e) and θ m (g m (k m )) = (e, b m ). Finally for all m we have g m = id since there are no commutators appearing in the decomposition of (g, 0).
Example 2.4. Assume that k m = 2 m and consider first the element (f , 0) of ∆ defined by (f , 0) := (0, -k 1 ) (a m δ 0 ) m , 0 (0, k 1 ). Now define the commutator and let us describe the values taken by g and the induced maps θ m (g m ) and g m (see Figure 2 for a representation of g). The only non-trivial commutator appearing in the values taken by g is g 1 (k 1 ) which is equal to a 1 b 1 a -1 1 b -1 1 . In other words g 0 is the identity, thus θ 0 = id. Moreover when m = 1 we have θ 1 = id and the only value of g 1 (x) different from e is g 1 (k 1 ) = a 1 b 1 a -1 1 b -1 1 (on a blue background in Figure 2). Finally if m > 1 then g m is the identity thus θ m = id and g m = id. Let us study the behaviour of this decomposition under product of lamp configurations. Combining Lemma 2.7 and Fact 2.9 of [BZ21], we get the following result.
In particular the sequence g = (g m ) m∈N is uniquely determined by g 0 and g m m∈N .
In the next subsection we are going to see that we actually need only a finite number of elements of the sequence (g m ) m∈N to characterise g.

Range and support
In this subsection we introduce the notion of range of an element (g, t) in ∆ and link it to the supports of the lamp configurations (g m ) m∈N .

Range
We denote by π 2 : ∆ → Z the projection on the second factor and for all n ∈ N denote by l(n) the integer such that k l(n) ≤ n < k l(n)+1 .

Definition 2.7
If w = s 1 . . . s m is a word over S ∆ we define its range as The range is a finite subinterval of Z. It represents the set of sites visited by the cursor.

Definition 2.8
The range of an element δ ∈ ∆ is defined as the diameter of a minimal range interval of a word over S ∆ representing δ.
In what follows we will consider elements that can be written as a word with range in an interval of the form [0, n], where n belongs to N. Therefore, when there is no ambiguity we will denote range(δ) this interval, namely range(δ) = [0, n].
Example 2.9. Let (g, 0) ∈ ∆ such that range(g, 0) = [0, 6], that is to say: the cursor can only visit sites between 0 and 6. Then the map g m can "write" elements of A m only on sites visited by the cursor, that is to say from 0 to 6, and it can write elements of B m only from k m to 6 + k m . Thus g 0 is supported on [0, 6], since k 0 = 0. Moreover, commutators (and hence elements of Γ m ) can only appear between k m and 6, thus supp(g m ) ⊆ [k m , 6].
In particular supp(g m ) is empty when k m > 6. Such a (g, 0) is represented in Figure 3 for k m = 2 m . Let us now recall a useful fact proved in [BZ21].
Example 2.11. Consider again (g, 0) ∈ ∆ such that range(g, 0) = [0, 6], which was illustrated in Figure 3. Since k 3 = 8 > 6, the element (g, 0) is uniquely determined by the data g 0 (that is to say, the values read in the bottom line) and the values of g i for i = 1, 2 (namely, the value taken in the blue area). Figure 4 represents the aforementioned characterizing data.

Relation between range and support
Recall that for all m ∈ N we can write g m ( To work with the Følner sequence we compute in Section 2.2.3 and deduce a Følner tiling shift from it, we will need to link the range of (g, t) in ∆ with the support of g 0 and the sequence of supports of (g m ) m∈N . This is what the following lemma formalises.

Lemma 2.12
Let n ∈ N and take (g, t) ∈ ∆. Then range(g, t) is included in [0, n] if and only if Proof. Let n ∈ N and first assume that range(g, t) ⊆ [0, n], that is to say: the cursor can only visit sites between 0 and n. Let (g, t) = l i=0 s i be a decomposition in a product of elements of S ∆ with range of minimal length. Let m ∈ N, then by definition of S ∆ , an element s i can "write" elements of A m only between 0 and n, and it can write elements of B m only between k m and n+k m . Thus g 0 is supported on [0, n], since k 0 = 0. And commutators can only appear between k m and n, hence supp(g m ) ⊆ [k m , n]. In particular if k m > n then g m ≡ e. Finally we obtain that t belongs to [0, n] by noting that t = π 2 l j=1 s j . Now let us prove the other way round. Consider m ∈ [1, l(n)] then g m (x) ∈ Γ m . It is therefore a product of conjugates of commutators of the form [a m , b m ], where a m ∈ A m and b m ∈ B m . Applying Example 2.4 with x instead of k 1 we can show that we can write [a m , b m ] at g m (x) without changing any other entry in g (see also Figure 2). In a similar way, we can write a conjugate of [a m , b m ] at g m (x) without changing any other entry in g. Finally writing (a 0 , b 0 ) at the entry g 0 (x) writes a m at g m (0) and b m at g m (k m ) (see also Figure 1). Therefore using Lemma 2.6 we can obtain (g, 0) by first considering the word in S ∆ that writes all the values of g 0 , then multiplying it on the left by a word that writes the value of g 1 , and continue this process to write all g m for m ≤ l(n).
Let us now check that the cursor remains in [0, n] when writing g 0 and g m . Take m ∈ [1, l(n)], then k m ≤ n and supp(g Combining what precedes with Lemma 2.6 and the hypothesis that t ∈ [0, n], we get that the cursor needs only to visit cites between [0, n] to write (g, t). Hence the lemma.

Følner sequence
In this subsection we describe a Følner sequence (F n ) n∈N for ∆. Recall that l(n) denotes the integer such that k l(n) ≤ n < k l(n)+1 .

From the isoperimetric profile to the group
We saw how to define a diagonal product from two sequences (k m ) m and (l m ) m . In this section we recall the definition given in [BZ21, Appendice B] of a Brieussel-Zheng group from its isoperimetric profile. We conclude with some useful results concerning the metric of these groups.

Definition of ∆
Recall that in the particular case of expanders (see Section 2.1.2) a Brieussel-Zheng group is uniquely determined by the sequences (k m ) m∈N and (l m ) m∈N (where l m corresponds to the diameter of Γ m ). Thus, starting from a prescribed function ρ, we will define sequences (k m ) m∈N and (l m ) m∈N such that the corresponding ∆ verifies I ∆ ρ • log. Let Equivalently this is the set of functions ζ satisfying ζ(1) = 1 and So let ρ ∈ C. Combining [BZ21, Proposition B.2 and Theorem 4.6] we can show the following result (remember that with our convention the isoperimetric profile considered in [BZ21] corresponds to 1/I ∆ ).
Proposition 2.14 Let κ, λ ≥ 2. For any ρ ∈ C there exists a subsequence (k m ) m∈N of (κ n ) n∈N and a subsequence (l m ) m∈N of (λ n ) n∈N such that the group ∆ defined in Section 2.1.2 verifies I ∆ (x) ρ • log.

Technical tools
We recall the intermediate functions defined in [BZ21, Appendix B] and some of their properties.
Let ρ ∈ C and let f such that ρ(x) = x/f(x). The construction of a group corresponding to the given isoperimetric profile ρ • log is based on the approximation of f by a piecewise linear functionf. For the quantification of orbit equivalence, many of our computations will usef and some of its properties. We recall below all the needed results, beginning with the definition off.

Lemma 2.16
Let ρ ∈ C and f such that ρ(x) = x/f(x). Let (k m ) and (l m ) given by Proposition 2.14 above and ∆ the corresponding diagonal product. The functionf defined bȳ Remark that bothf andρ belong to C. In particular they verify Equation (2.2), which is only true when c and x are greater than 1. When c < 1 we get the following inequality.

Metric
We recall here some useful material about the metric of ∆ and refer to [BZ21, Section 2.2] for more details. First, let (x) + := max{x, 0}. The following proposition sums up [BZ21, Lemma 2.13, Proposition 2.14].

folner tiling shifts
We start by recalling some material of [DKLMT22] about Følner tiling shifts and then construct such a tiling for diagonal products.

Følner tiling shifts
The tools we are going to use to build orbit equivalence are Følner tiling shifts 2 . These sequences lead to Følner sequences defined recursively: the term of rank (n+1) is composed of a finite number of translates of the n-th term of the sequence.
Definition 3.1 Let G be an amenable group and (Σ n ) n∈N be a sequence of finite subsets of G. Define by induction the sequence (T n ) n∈N by T 0 := Σ 0 and T n+1 := T n Σ n+1 . We say that (Σ n ) n∈N is a (left) Følner tiling shift if • (T n ) n∈N is a left Følner sequence, viz. lim n→∞ |gT n \T n |/|T n | = 0 for all g ∈ G; • T n+1 = σ∈Σ n+1 σT n . We call Σ n the set of shifts and (T n ) n∈N the tiles.
We can also consider right Følner tiling shifts, that is to say sequences (Σ n ) n such that T n+1 := Σ n+1 T n defines a right Følner sequence.

Definition 3.2
Let S be a generating part of G. We say that (Σ n ) n∈N is a (R n , ε n )-Folner tiling shift if for all n we have diam (T n ) ≤ R n , |sT n \T n | ≤ ε n |T n | (∀s ∈ S).
Example 3.4. If G = (Z/2Z) Z then the sequence (Σ n ) n∈N defined by is a right (3 · 2 n , 2 -n )-Følner tiling shift. Moreover the tiling (T n ) n∈N thus defined verifies In [DKLMT22] the authors used Følner tiling shifts to build an explicit orbit equivalence coupling between two amenable groups and quantify its integrability. Indeed if G admits a Følner tiling shift (Σ n ) n∈N then we can define X := n∈N Σ n and endow it with an action of G. Up to measure zero, two elements of X will be in the same orbit under that action if and only if they differ by a finite number of indices. The equivalence relation thus induced is called the cofinite equivalence relation. Now if G admits a Følner tiling shift (Σ n ) n∈N verifying |Σ n | = |Σ n | for all integer n, then there exists a natural bijection between X and X := n∈N Σ n which preserves the cofinite equivalence relation. That is to say G and H are orbit equivalent. Furthermore they showed that if we know the diameter and the ratio of elements in the boundary of each tile, then we can deduce the integrability of the coupling. This is what the following proposition sums up. Let G and G be two discrete amenable groups and let (Σ n ) n be an (ε n , R n )-Følner tiling shift for G and (Σ n ) n be an (ε n , R n )-Følner tiling shift for G .
If |Σ n | = |Σ n |, then the groups are orbit equivalent over X = n∈N Σ n . Moreover if ϕ : R + → R + is a non-decreasing map such that the sequence ϕ(2R n ) (ε n-1ε n ) n∈N is summable, then the coupling from G to G is (ϕ, L 0 )-integrable.
Using this tiling technique and the above theorem, Delabie et al. [DKLMT22] obtained the first point of Examples 1.4 and the two following quantifications.
Example 3.6. For all n and m there exists an orbit equivalence coupling from Z m to Z n which is (ϕ ε , ψ )-integrable for every ε > 0 where Remark that in particular for all p < n/m and q < m/n there exists a (L p , L q )-orbit equivalence coupling from Z m to Z n .
Example 3.7. Let m ≥ 2. There exists an orbit equivalence coupling from Z to Z/mZ Z that is (exp, ϕ ε )-integrable for all ε > 0 where ϕ ε (x) = log(x) log(log(x)) 1+ε . Note that the above example corresponds to the case when ρ(x) = x in our Theorem 1.7.

Følner tiling shifts of diagonal products
Let (k m ) m and (l m ) m be two sequences verifying the conditions of (H) and consider ∆ the associated diagonal product (see Section 2). We define below a Følner tiling shift for ∆. Our goal is to obtain a tiling verifying T n = F κ n . After defining the shifts sets Σ n we prove that the sequence (Σ n ) n∈N is actually a Følner tiling shift. Finally we make this last statement precise by computing (R n ) n∈N and (ε n ) n∈N such that (Σ n ) n∈N is a (R n , ε n )-Følner tiling shift (see Definition 3.1).

Definition of the shifts
For any n ∈ N, let L(n) = l(κ n -1), that is to say L(n) is the integer such that k L(n) ≤ κ n -1 < k L(n)+1 . For example if k n := κ n for all n ∈ N, then L(n) = n -1.
Before defining our sequence (Σ n ) n∈N , let us show some practical results on L. First remark that since (k n ) n∈N is a subsequence of (κ n ) n∈N , it verifies k n ≥ κ n for all n ∈ N. Thus L(n) ≤ n and k L(n) < κ n ≤ k L(n)+1 .
Proof. Recall that by definition L(m) = max {i ∈ N | k i ≤ κ m -1} for all m ∈ N.
Finally if L(n + 1) = L(n) + 1 then by definition of L But (k m ) m∈N is a subsequence of κ m thus the above inequality implies k L(n+1) = κ n . Now, let us define the shifts. First let Σ 0 := F 0 , then if n ≥ 0 we distinguish two cases depending on whether L(n + 1) = L(n) or L(n + 1) = L(n) + 1 and in both cases we split the set of shifts Σ n+1 in κ parts.

Tiling
Recall that (F n ) n∈N denotes the Følner sequence of ∆ defined in Proposition 2.13. The aim of this section is to show the theorem below.
Theorem 3.9 The sequence (Σ n ) n∈N defined in Section 3.2.1 is a Følner tiling shift of ∆. Lemma 3.10 The sequence (T n ) n∈N defined by T 0 := F 0 and T n+1 := Σ n+1 T n for all n > 0 verifies ∀n ∈ N T n = F κ n .
Let us discuss the idea of the proof. We proceed by induction and use a double inclusion argument to prove the induction step. To show that Σ n+1 T n is included in F κ n+1 we rely on Lemma 2.12, that is to say we verify that every element of Σ n+1 T n has range included in [0, κ n+1 -1]. For the reversed inclusion we consider an element (h, t) of F κ n+1 and make the elements (g, jκ n ) of Σ n+1 and (f , t ) of T n explicit such that (h, t) = (g, jκ n )(f , t ).
Mind the involved maps here: we study the values of g m and f m instead of the "derived" functions g m , f m usually considered.
Proof of the lemma. The assertion is true for T 0 . Now let n ≥ 0 and assume that T n = F κ n . We show the induction step by double inclusion.

Second Inclusion
Let us show that F κ n+1 is contained in Σ n+1 T n . So take (h, t) in F κ n+1 . We want to define f , t ∈ T n and (g, jκ n ) ∈ Σ n+1 such that (g, jκ n ) f , t = (h, t). First remark that t < κ n+1 since (h, t) belongs to F κ n+1 . Thus there exists t 0 , . . . , t n in [0, κ -1] such that t = n i=0 t i κ i . Let j = t n and t = n-1 i=0 t i κ i . Then j does belong to [0, κ -1] and t to [0, κ n -1]. We now have to define f and g such that (g m f m (· -jκ n )) m , t + jκ n = h, t .
We refer to Figure 6 for an illustration of the different supports. Let One can verify immediately that g 0 f 0 (· -jκ n ) = h 0 . Then take m ∈ [1, L(n)] and let Now if L(n+1) = L(n)+1 then k L(n+1) ≥ κ n and in that case define g L(n+1) = h L(n+1) .
Finally let f L(n+1) ≡ e and if m > L(n + 1) let g m ≡ e ≡ f m .
With the above definitions f and g are uniquely defined. Moreover, by definition (g, jκ n ) belongs to Σ j n+1 and by Lemma 2.12 we have range(f , t) ⊆ [0, κ n -1] thus (f , t ) belongs to T n . Now, using Lemma 2.6 we verify that g m f m (· -jκ n ) = h m thus (h, t) ∈ Σ n+1 T n .
Hence, combining the first and second inclusion we get F κ n+1 = T n .
We now know that (T n ) n∈N is a Følner sequence. To prove Theorem 3.9 we have to show that (Σ n ) n∈N a Følner tiling shift.
Proof of Theorem 3.9. The sequence (T n ) n∈N is a Følner sequence, by the last lemma. Thus we only have to show that for all σ =σ ∈ Σ n+1 , σT n ∩σT n = ∅. So let us denote by (h, t) an element of σT n ∩σT n . We distinguish two cases.
If x ∈ [jκ n , jκ n + k m -1] then using Lemma 2.6 and the fact that on that subinterval f 0 =f 0 , we get Hence by Equation (3.1) we get g m (x) =g m (x). If x belongs to [jκ n + k m , (j + 1)κ n -1] then g m (x) =g m (x) = e and thus Equation (3.1) implies that f m (x -jκ n ) =f m (x -jκ n ), that is to say f m andf m coincide on [k m , κ n -1].
Finally if x ∈ [(j + 1)κ n , (j + 1)κ n + k m -1] then using Lemma 2.6 and the fact that f 0 =f 0 on that subinterval, we get Hence by Equation

Diameter and boundary
Let us now quantify our shifts sequence.

Proposition 3.11
The sequence (Σ n ) n∈N defined in Section 3.2.1 is a (R n , ε n )-Følner tiling shift where R n = C R κ n l L(n) ε n = 2 κ n , for some strictly positive constant C R .
First we prove the following lemma.

Lemma 3.12
There exists C R > 0 depending only on ∆ such that diam (F n ) ≤ C R nl l(n-1) for all n ∈ N.
To show this result, we use Proposition 2.20.
Proof. Let n ∈ N and (f , t) ∈ F n . First, take m ≤ l(n -1) and let us bound E m by above.
Thus, applying the second part of Proposition 2.20 we get But if m ≤ l(n -1) then k m ≤ n -1 ≤ n thus we can bound |(f m , t)| ∆m by above by 9n(3l m +1). Now remark that l (range(f , t)) ≤ l(n-1). Thus, using the preceding inequality and the first part of Proposition 2.20, we get Finally, since l m is a subsequence of a geometric sequence, there exists C l > 0 such that l(n-1) m=0 (3l m + 1) ≤ C l l l(n-1) . Denoting C R := 4500C l we get the lemma.
Let us now show the wanted proposition.
Proof of Proposition 3.11. First remark that by the proof of Proposition 2.13 we have Now by Lemma 3.12 we have diam (T n ) = diam (F κ n ) ≤ C R κ n l L(n) .

coupling with
Our aim in this section is to show Theorem 1.7. What we actually show is that a diagonal product ∆ admits a coupling with Z satisfying Theorem 1.7. We start by defining a Følner tiling shift for Z in Section 4.1. We compute in Section 4.2 an estimate of the diameter of such tiles, namely the cardinal |T n |. We conclude by showing the integrability of the coupling using the criterion given by Theorem 3.5. And then show that ∆ thus considered satisfies Theorem 1.7.

Tiles for Z
We will denote by (Σ n ) n∈N a Følner tiling shift of Z and by (T n ) n the corresponding tiles. Consider (Σ n ) n and (T n ) n as defined in Section 3.2.1 and Lemma 3.10 respectively. In order to use Theorem 3.5 to get an orbit equivalence coupling between Z and ∆ we need Σ n+1 and Σ n+1 to have the same number of elements. We thus define Σ 0 = 0, |T 0 | -1 ∀n ∈ N Σ n+1 := 0, |T n |, 2|T n |, . . . , (|Σ n+1 | -1) |T n | . (4.1) It induces a sequence (T n ) n∈N defined by T 0 = Σ 0 and T n+1 = Σ n+1 T n for all n ≥ 0. We are going to prove that (Σ n ) n∈N is a Følner tiling shift for Z.
Moreover the induced sequence (T n ) n∈N verifies T n = [0, |T n | -1] for all n ∈ N.
Proof. Let (Σ n ) n∈N be as defined by Equation (4.1) and recall that the induced tiling (T n ) n∈N is the sequence defined by T 0 := Σ 0 and T n+1 = Σ n+1 T n for all n ∈ N. One can easily prove that for all n ≥ 0 It is now immediate to check that diam T n = |T n | and |∂T n |/|T n | = 2/|T n |. Furthermore note that if σ, σ ∈ Σ n+1 such that σ = σ then d Z (σ, σ ) ≥ |T n | = diam T n . Thus for such σ and σ we get σT n ∩ σ T n = ∅. Therefore (Σ n ) n∈N is a Følner tiling shift and the proposition follows from the above quantifications on T n .

Estimates: diameter and boundary
The integrability of the coupling between Z and ∆ depends on (R n , ε n ) and (R n , ε n ) but by the above proposition, that last couple depends on the value of the cardinality of the tiles (T n ) n∈N . The aim of this section is to give estimates of |T n | involving only terms of (k m ) m∈N and (l m ) m∈N . First let us make the value of |T n | precise.

Lemma 4.2
The sequence (T n ) n defined in Theorem 3.9 verifies

Proposition 4.3
There exists two constants C 2 , C 3 > 0 such that for all n ∈ N, C 2 κ n-1 l L(n) ≤ ln |T n | ≤ C 3 κ n l L(n) .
Before showing the above proposition let us give an estimate of the right factor of the expression of |T n |.

Lemma 4.4
There exists two constants C 1 , C 2 > 0 such that for all n ∈ N, Proof. Recall that by Equation (2.1) there exists c 1 , c 2 > 0 such that, for all m c 1 l m -c 2 ≤ ln |Γ m | ≤ c 1 l m + c 2 .
Since Γ m ≤ Γ m we thus have But we can bound κ n -k m from above by κ n and since (l m ) m∈N is a subsequence of a sequence having geometric growth, the sum L(n) m=1 (c 1 l m + c 2 ) is bounded from above by its last term up to a multiplicative constant. That is to say: there exists C 1 > 0 such that Hence the upper bound. Now, using that [Γ m : Bounding the sum from below by its last term and using once more Equation (2.1), we get ≥ C 2 (κ n -k L(n) )l L(n) , for some C 2 > 0. We get the wanted inequality by noting that κ n -k L(n) ≥ κ n-1 .
Proof of Proposition 4.3. Applying Lemma 4.4 to the cardinal of T n given by Lemma 4.2 we obtain that there exists C 3 > 0 such that ln |T n | ≤ C 3 κ n l L(n) . Hence the upper bound. The minoration comes imediately from Lemma 4.4.
Equipped with these bounds on |T n | we can now show the wanted integrability for the coupling.

Integrability of the coupling
We will show that ∆ is the group satisfying Theorem 1.7, but first let us quantify the integrability of the orbit equivalence coupling with Z induced by the Følner tiling shifts we built. Recall that C denotes the set of non-decreasing functions ρ : [1, +∞[→ [1, +∞[ such that x/ρ(x) is non-decreasing.

Theorem 4.5
Let ρ ∈ C and take ∆ to be the Brieussel-Zheng diagonal product defined from ρ. Let ε > 0 and Ψ := exp •ρ and let There exists an orbit equivalence coupling from ∆ to Z that is (ϕ ε , Ψ)-integrable.
Let us discuss the strategy of the proof. The demonstration is based on Theorem 3.5, thus we first prove that (Ψ(2R n )ε n-1 ) n is summable and then that (ϕ ε (2R n )ε n-1 ) n is. In both cases we use Proposition 4.3 to get upper bounds. So far, we have the following quantifications.
R n = C R κ n l L(n) R n = |T n | ε n =2κ -n ε n = 2/|T n | Proof of Theorem 4.5. Let ρ ∈ C and take ∆ to be the diagonal product defined from ρ as described in Section 2.3. To begin, let us recall some preliminary results about ρ. Remember that ρ ρ wherē ρ is defined below Equation (2.3). By definition of L(n) we have k L(n) l L(n) ≤ κ n l L(n) ≤ k L(n)+1 l L(n) , thus by Equation (2.3)ρ (κ n l L(n) ) = κ n . (4.3) Now let us show that the coupling from Z to ∆ is Ψ-integrable. To do so we prove that Ψ(2R n )ε n-1 is summable. First note that by Proposition 4.3 we have the following lower bound on |T n-1 | |T n-1 | ≥ exp C 2 κ n-2 l L(n-1) . (4.4) Moreover recall that R n = C R κ n l L(n) and ε n-1 = 2/|T n-1 | thus by the inequality above Ψ(2R n )ε n-1 = exp ρ(2C R κ n l L(n) ) 2 |T n-1 | , ≤ 2 exp ρ 2C R κ n l L(n) -C 2 κ n-2 l L(n-1) .
Which is summable by choice of c ϕ .
Remark 4.7. We can verify that the integrability obtained for the coupling from ∆ to Z is "almost" optimal. Indeed if the coupling from ∆ to Z is ϕ-integrable, then by Theorem 1.5 we have ϕ • I Z I ∆ where we recall that I Z (n) n and I ∆ (n) ρ • ln(n). Thus using the inequality above, we get ϕ(n) ρ • ln(n). Hence the quantification of Theorem 4.5 is optimal up to a logarithmic factor.
It is now easy to prove our first main theorem.
Proof of Theorem 1.7. Let ρ ∈ C and ∆ to be the group defined in Proposition 2.14. By the aforementioned proposition it verifies I ∆ ρ • log. Moreover by Theorem 4.5 there exists an orbit equivalence coupling from ∆ and Z that is (ϕ ε , exp •ρ)-integrable for all ε > 0.
To prove Corollary 1.8 we use the composition of couplings introduced in [DKLMT22]. We recall below the proposition concerning the integrability of this composition and refer to [DKLMT22, Sections 2.3 and 2.5] for more details on the construction of the corresponding coupling.

conclusion and open problems
Let us conclude with some questions and remarks.

Optimality and coupling building techniques
The tiling technique -though inspiring-is not always usable to get orbit equivalence couplings. Indeed the condition that the two Følner tiling shifts must have at each step the same cardinality is very restrictive. Furthemore this technique does not seem to produce couplings with the best quantification: wether it is our coupling with Z or the one built in [DKLMT22] (Examples 3.6 and 3.7) the integrability is always optimal up to a logarithmic factor. One can thus ask: is the optimal integrability reachable? Is the logarithmic error due to the building technique?

Inverse problem
We studied here the inverse problem for the group of integers (Question 1.6) but one can also ask the same question for other groups than Z.
Question 5.1. Given a function ϕ and a group H is there a group G such that there exists a (ϕ, L 0 )-measure equivalent from G to H? Can G be chosen such that ϕ • I H I G ?
In [Esc22] we answer this question when H is a diagonal product, in particular H can be a lamplighter group. This coupling is obtained with another building technique than the tiling process and the integrability is optimal, answering the questions of Section 5.1 positively.