Extensions of Schreiber’s theorem on discrete approximate subgroups in <span class="mathjax-formula">$\protect \mathbb{R}^d$</span>

Extensions of Schreiber’s theorem on discrete approximate subgroups inR Tome 6 (2019), p. 149-162.

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EXTENSIONS OF SCHREIBER’S THEOREM ON DISCRETE APPROXIMATE SUBGROUPS IN R

by Alexander Fish

Abstract. — In this paper we give an alternative proof of Schreiber’s theorem which says that an infinite discrete approximate subgroup inR^dis relatively dense around a subspace. We also deduce from Schreiber’s theorem two new results. The first one says that any infinite discrete approximate subgroup inR^dis a restriction of a Meyer set to a thickening of a linear subspace inR^d, and the second one provides an extension of Schreiber’s theorem to the case of the Heisenberg group.

Résumé(Extensions du théorème de Schreiber sur les sous-groupes approximatifs discrets deR^d) Dans cet article, nous donnons une autre démonstration du théorème de Schreiber : un sous- groupe approximatif discret infini deR^dest relativement dense au voisinage d’un sous-espace.

Nous déduisons aussi du théorème de Schreiber deux nouveaux résultats : le premier affirme qu’un sous-groupe approximatif discret infini deR^dest la restriction d’un ensemble de Meyer à un épaississement d’un sous-espace linéaire de R^d, et le second propose une extension du théorème de Schreiber au cas du groupe de Heisenberg.

Contents

1. Introduction. . . 149

2. Main results. . . 151

3. Discrete approximate subgroups inR^d. . . 153

4. Discrete approximate subgroups in the Heisenberg group. . . 160

References. . . 162

1. Introduction

In this paper we study approximate subgroups. Recall that for a group H, a set Λ⊂H is called an approximate subgroupif there exists a finite setF ⊂H such that Λ⁻¹Λ⊂FΛ, whereΛ⁻¹ ={λ⁻¹|λ∈Λ}. In the case whereH is non-commutative, we will also assume as a part of the definition that Λ contains the identity element ofH andΛis symmetric:

– e_H∈Λ, – Λ⁻¹= Λ.

2010Mathematics Subject Classification. — 11B30, 52C23.

Keywords. — Approximate groups, approximate lattices, Meyer sets.

Any finite set in a groupHis contained in an approximate subgroup. An interesting question of classification of approximate subgroups arises if we control the cardinality ofF, while the cardinality ofΛis finite but much larger than of the set of translatesF, and in this case we say thatΛ has a small doubling. The classification of finite sets having small doubling for the ambient group H =Z has been obtained by Freiman in his seminal work [3]. These results have been eventually extended to all abelian groups by Green and Ruzsa [4], and to arbitrary ambient groups by Hrushovski [5], and by Breuillard, Green and Tao [2].

We will investigate here infinite discrete approximate subgroups inR^d and in the Heisenberg group. Infinite discrete relatively dense approximate subgroups in R^d, Meyer sets, have been studied extensively by Meyer [7], Lagarias [6], Moody [8] and many others. It has been proved by Meyer [7] that a discrete relatively dense approx- imate subgroup inR^d is a subset of a model (cut and project) set [8]. Thus, despite a possible aperiodicity of Meyer sets, they all arise from lattices in (possibly) much higher dimensional spaces. Very recently, there was a spark of interest in the exten- sion of Meyer theory beyond the abelian case. A foundational work of Björklund and Hartnick [1] introduced the notion of an approximate lattice within Lie groups. The approximate lattices behave similarly to genuine lattices, and therefore, they are a good analog of Meyer sets in the non-abelian case.

The paper addresses a natural question of what kind of structure possesses an in- finite discrete approximate subgroup Λ in R^d (or in the Heisenberg group) which is not relatively dense in the whole space. It has been almost forgotten by the math- ematical community, that Schreiber in his thesis in 1972 [9] proved that in the real caseΛhas to be relatively dense around a subspace, see definition 2.1. We provide an alternative, more geometric, proof of Schreiber’s result. We also extend his theorem and show that any discrete infinite approximate subgroup inR^dis a subset of a Meyer set. In addition, we extend Schreiber’s theorem to the case where the ambient space is the Heisenberg group.

Remark1.1. — The first draft of the paper dealt only with the real case (and had a different title). It was not known to the author that Theorem 2.2 was already proved by Schreiber. The author thanks Simon Machado for providing the reference to Schreiber’s thesis.

Remark1.2. — After the first draft of the paper was released, Simon Machado has obtained the analog of Theorem 2.7 to all connected real nilpotent groups.

Acknowledgment. — The author is grateful to American Institute of Mathematics (AIM) and the organisers of the workshop on “Nonstandard methods in combinatorial number theory” at AIM, where this project has been initiated. We also thank Terrence Tao who suggested the statement of Theorem 2.4 in the cased= 2. We would like also to thank Benji Weiss for fruitful discussions and anonymous referees that made very invaluable comments on the first draft of the paper, and, in particular, asked a question that led to Theorem 2.3. The paper has been influenced by Michael Björklund, and,

particularly, by his series of lectures on quasi-crystals given at Sydney University in April 2016. We thank Michael for sharing his mathematical ideas with us. Finally, the author thanks Simon Machado for sharing with him his insights on the topic.

2. Main results

We will always assume that the underlying groupH possesses a leftH-invariant metricd_H, and for anyr >0 andh∈H we will denote by

Br(h) ={g∈H |dH(g, h)6r}

the ball of radius r around h. We will call a set Λ ⊂ H discrete if for every point

` ∈ Λ there exists δ = δ(`) such that Bδ(`)∩Λ = {`}. It is well known, that if Λ ⊂H is a discrete approximate subgroup, then Λ is uniformly discrete, i.e., there exists δ >0such that for all`∈Λwe haveBδ(`)∩Λ ={`}. Indeed, Λis uniformly discrete if and only ifΛ⁻¹Λ does not contain identityeH as an accumulation point.

Since e_H ∈ Λ⁻¹Λ, and Λ⁻¹Λ is discrete, it follows that e_H is not an accumulation point of Λ⁻¹Λ, and thereforeΛis uniformly discrete. We will call Λ relatively dense (or R-relatively dense) if there exists R > 0 such that for every h ∈ H we have BR(h)∩Λ6=∅.

The following notion will play a key role in our paper.

Definition2.1. — LetH⁰ be a subgroup of the groupH. We will say thatΛ⊂H is relatively dense aroundH⁰ if there existsR >0such that:

– For everyh∈H⁰the ballBR(h)of radiusRand centrehintersects non-triviallyΛ.

– TheR-neighbourhood ofH⁰ in H containsΛ, i.e., Λ⊂ S

h∈H⁰

BR(h).

2.1. Discrete approximate subgroups in R^d. — In this paper we give an alterna- tive proof of Schreiber’s theorem [9] that discrete approximate subgroups inR^d are relatively dense around some subspace.

Theorem2.2(Schreiber, 1972). — Let Λ ⊂ R^d be an infinite discrete approximate subgroup. Then there exists a linear subspace L⊂R^d such that Λ is relatively dense aroundL.

As a corollary of Theorem 2.2 we obtain a complete characterisation of infinite approximate subgroups inR^d in terms of Meyer sets. Recall, that a set inΛ⊂R^d is a Meyer set if

– Λis discrete and relatively dense in R^d,

– There exists a finite setF ⊂R^d such thatΛ−Λ⊂Λ +F.

Theorem2.3. — A set Λ ⊂R^d is an infinite discrete approximate subgroup if and only if there exists a Meyer set Λ⁰⊂R^d, a subspaceL⊂R^d andR >0 such that

Λ = Λ⁰∩(L+B_R(0_Rd)), andΛ⁰ isR/2-relatively dense in R^d.

As another corollary of Theorem 2.2, we obtain a complete characterisation of infinite approximate subgroups inZ^d.

Theorem2.4. — Let Λ be a subset in Z^d. The setΛ is an infinite approximate sub- group if and only if there exists a linear subspace L ⊂ R^d such that Λ is relatively dense around L.

A third application of Theorem 2.2 is that any discrete approximate subgroup inR^d is “very close” to being a Meyer set on a subspace ofR^d. More precisely, we prove the following result.

Proposition2.5. — Let Λ⊂R^d be an infinite discrete approximate subgroup. Then there exist a subspaceL⊂R^d andR >0 such that:

– The orthogonal projection ΛL of Λ on the subspace Lis a Meyer set in L, i.e., Λ_L is discrete relatively dense approximate subgroup inL,

– Λ⊂Λ_L+B_R(0_Rd).

The following example shows that an infinite discrete approximate subgroupΛ is not necessarily subset of finitely many translates ofΛL.

Example 2.6. — Let L = Span((1,√

3)) ⊂ R², and let Λ ⊂ R² be the set of all (m, n)∈Z² such thatdist((m, n), L)61. Then Λ is discrete since it is a subset of the integer lattice, andΛ−Λ⊂Λ +F for a finite set⁽¹⁾F ⊂Z². But the orthogonal projection of Λ on the orthogonal complement ofL in R² is infinite, since the slope ofLis irrational. This implies that for any finite setF ⊂R² we have

Λ6⊂ΛL+F,

whereΛL is the orthogonal projection ofΛonto the lineL.

2.2. Discrete approximate subgroups in the Heisenberg group. — For any n >1 we define the Heisenberg group H2n+1 by the following procedure. Assume that ω:R²ⁿ×R²ⁿ →R is a symplectic form, i.e., ω is a bilinear, anti-symmetric and non-degenerate form. Then the Heisenberg group H2n+1 =R²ⁿoωRis defined by H2n+1={(v, z)|v∈R²ⁿ, z∈R}, and the multiplication is given by

(v1, z1)·(v2, z2) =

v1+v2, z1+z2+1

2ω(v1, v2) .

We will denote by V the symplectic space R²ⁿ, and by Z the abelian subgroup Z ={(0, z)|z∈R}. The subgroupZ is the centre ofH2n+1. The Heisenberg group H2n+1 is 2-step nilpotent. Indeed, for any two elementsh1= (v1, z1), h2 = (v2, z2)∈ H_2n+1, the commutator ofh₁andh₂ satisfies

(1) [h₁, h₂] = (0, ω(v₁, v₂)).

(1)We can takeF=Z²∩B2(0_R2).

It is easy to see that the Heisenberg group can be equipped with a left invariant metric.

In the topology defined by this metric, the sequence (vn, zn)in H2n+1 converges to (v, z)if and only ifv_n→vin V andz_n→z inZ.

We extend Schreiber’s theorem to the Heisenberg case.

Theorem2.7. — Let Λ⊂H2n+1 be an infinite discrete approximate subgroup. Then there exists a connected non-trivial subgroup H⁰ in H2n+1 such that Λ is relatively dense around H⁰. Moreover, ifH⁰ is non-abelian, then the projection of Λ ontoV is discrete.

Let us denote byπV the projection fromH2n+1 ontoV, i.e.,πV((v, z)) =vfor any (v, z)∈H2n+1. The following example shows that we cannot improve the statement of the theorem.

Example 2.8. — Let Λ = {(m√ 5 +n√

3,0), m) | m, n ∈ Z}. Then Λ is a discrete subgroup in H3. It is clear that πV(Λ) is dense within L = R× {0} ⊂ V, and therefore it is non-discrete.

It is easy to see that the projection on theZ-coordinate of a discrete approximate subgroup is not necessarily discrete. Indeed, we can find lattices inH₃ with a dense set of theZ-coordinates.

Example2.9. — It is easy to see that Λ =n

(m, n), m√

5 +¹₂Z

|m, n∈Z o

is a discrete co-compact subgroup in H3 (equipped with the determinant on R² as the symplectic form). But the projection ofΛonZ is everywhere dense.

By the methods similar to the ones used to prove Theorem 2.7, we prove also the following claim.

Proposition2.10. — LetΛ⊂H2n+1 be a discrete approximate subgroup. IfπV(Λ)is relatively dense in V, then Λ is an approximate lattice, i.e., Λ is relatively dense in H_2n+1.

The analog of Theorem 2.3 is not possible in the Heisenberg group:

Proposition 2.11. — Let Λ = {(m√ 5 +n√

3,0), m) | m, n ∈ Z}. Then there is no approximate lattice (discrete relatively dense approximate subgroup) in H3 which containsΛ.

3. Discrete approximate subgroups inR^d

Let Λ ⊂ R^d be an approximate subgroup. By translating Λ if necessary, we can assume that 0_Rd ∈Λ. Denote K = diam(F). Since, for any two `1, `2 ∈Λ we have

`1−`2∈Λ +F, this implies that there exists`∈Λwith`1−`2∈BK(`). If, we take

`₁= 0, we obtain the following property ofΛ:

(A) for every`∈Λ there exists`⁰ ∈BK(−`)∩Λ.

By use of the property (A), if we take`2∈Λ, then there exists `3 ∈Λ such that

`3 ∈BK(−`2). Also for every`1 ∈Λ,there exists`4∈Λsuch that `1−`3∈BK(`4).

Finally, this implies that `<sub>1</sub>+`₂ ∈B_2K(`₄). Thus, the following property holds for any approximate subgroupΛcontaining the neutral element:

(B) for any`1, `2∈Λthere exists`<sup>0</sup>∈B2K(`1+`2)∩Λ.

We will call the property (A) the almost symmetry, and (B) the almost doubling. We start with an easy observation which proves Theorem 2.2 in the cased= 1.

Proposition3.1. — LetΛ⊂Rbe an infinite discrete approximate subgroup. ThenΛ is relatively dense.

Proof. — Assume thatΛ⊂Ris an infinite approximate subgroup. Take`∈Λ with

` >3K(which exists by uniform discreteness ofΛ). By the almost doubling property there exists`2 ∈Λ with`2∈[2`−2K,2`+ 2K]⊂[`+K,2`+ 2K]. Similarly, there exists`3∈Λ∩B2K(`2+`). Therefore,`3∈[`2+K, `2+`+2K]. Assume that we already constructed`1 =`, `2, . . . , `n ∈Λsatisfying that `m+K6`m+1 6`m+`+ 2K for m= 1, . . . , n−1. Then there exists`<sub>n+1</sub>∈Λ∩[`_n+`−2K, `_n+`+ 2K]. Therefore, we constructed an increasing sequence in Λ∩R+ with bounded gaps. By almost symmetry property ofΛ, we also have inΛthe elements{−`⁰,−`<sup>0</sup><sub>2</sub>, . . . ,−`⁰_n, . . .}with

`<sup>0</sup> ∈B<sub>K</sub>(−`). This finishes the proof of the Proposition.

A higher-dimensional case is much more subtle. An important role in the proof of Theorem 2.2 will play the set of asymptotic directions of the points inΛ.

Definition3.2. — LetΛ⊂R^d be a uniformly discrete infinite set. We call D(Λ) =

u∈S^d−1| there exists(`<sub>n</sub>)∈Λwith`_n/k`nk →uandk`nk → ∞ theset of asymptotic directionsofΛ.

It is easy to see that D(Λ) is non-empty closed set. It will be very convenient to us to introduce the subspace generated by D(Λ). Let L⊂R^d be the smallest linear subspace with the property thatD(Λ)⊂L. In other words, we have

L= Span(D(Λ)).

The next lemma is an important ingredient in the proof of Theorem 2.2.

Lemma3.3. — Assume thatΛ is an infinite discrete approximate subgroup. LetL= Span(D(Λ)) be a proper subspace inR^d. Then there existsR >0 (R= 3·diam(F)) such that

Λ⊂ S

x∈L

B_R(x).

Proof. — LetΛ∈R^d be an infinite discrete approximate subgroup, i.e., there exists a finite set F ⊂R^d with Λ−Λ⊂Λ +F. DenoteK = diam(F). For any ε >0 and anyu∈S^d−1we define the cone

Vε(u) ={tv|t >0, v∈S^d−1 withhv, ui>1−ε}.

Let us takeR= 3K. We claim that Λ⊂ S

x∈L

BR(x).

Indeed, if there exists ` ∈ Λ such that ` 6∈ S

x∈LBR(x), let us define u = `/k`k and1−ε=p

k`k<sup>2</sup>−5K<sup>2</sup>/k`k. Then we construct a sequence`<sub>1</sub>, `₂, `₃, . . . in Λwith

`<sub>n</sub> → ∞and`_n∈V_ε(u). Since, clearly, we have Vε(u)∩L={0_Rd}, this will imply the contradiction.

The construction is the same as in the proof of Proposition 3.1. Let us define`1=`.

We find`2∈B2K(`1+`)∩Λ. The following calculation guarantees that`2∈Vε(u):

`<sub>2</sub>/k`₂k, u

> 2k`k

p4k`k<sup>2</sup>+ 4K<sup>2</sup> = k`k

pk`k²+K² >1−ε.

Also, it is clear that k`2k>k`1k+K. Assume that we constructed a finite sequence

`<sub>1</sub>, `₂, . . . , `<sub>n</sub>∈Λwithk`_m+1k>k`<sub>m</sub>k+K,m= 1, . . . , n−1, and`₁, `<sub>2</sub>, . . . , `_n∈V_ε(u).

Then there exists`n+1∈B2K(`n+`)∩Λ. Clearly, we have k`n+1k>k`nk+K.

Finally, for any vectorv∈V_ε(u)we have

B2K(v+`)⊂Vε(u).

This will guarantee that`_n+1∈V_ε(u). Indeed, if a vectorv∈V_ε(u), thenv+V_ε(u)⊂ Vε(u), and therefore we have:

dist(v+`, ∂Vε(u))>dist(v+`, ∂(v+Vε(u)))

= dist(`, ∂(Vε(u))) =k`k(1−ε) =p

k`k²−5K²>2K.

Our next step in the proof of Theorem 2.2 is to construct a system of “basis”

vectors forΛ. LetL= Span(D(Λ)), and letRsatisfy

(2) Λ⊂ S

x∈L

BR(x).

Assume that dim(L) = k, where 1 6 k 6 d, and denote K = diam(F). By the definition of the set of asymptotic directions D(Λ), there exists ε >0 such that for everyM >0 there existkelements`1, . . . , `k∈Λsatisfying the following properties:

– (ε-well spreadness) For all 1 6 i 6 k, anyv_i ∈ B_2K(`<sub>i</sub>), and v<sub>j</sub> ∈ B<sub>2K</sub>(ε<sub>j</sub>`_j), j6=i, εj∈ {−1,1}, let us denote byγi the angle betweenvi and the subspace

Vⁱ= Span{v1, . . . , vi−1, vi+1, . . . , vk}.

Then we require:

ε6γi6π−ε, – (no short vectors) For every16i6kwe have

k`ik>M.

By almost symmetry ofΛ, we can also find the “reflected” vectors {`<sup>0</sup><sub>1</sub>, . . . , `⁰_k} ⊂Λ which satisfy the property

`<sup>0</sup><sub>i</sub> ∈BK(−`i), i= 1, . . . , k.

Let us denote F = {`<sub>1</sub>, . . . , `_k, `<sup>0</sup><sub>1</sub>, . . . , `⁰_k}. By Lemma 3.3 there exists R > 0 such thatΛ⊂LR, whereLR=S

x∈LBR(x)is theR-thickening of the subspaceL. Let us assume thatR >K. Finally, for any choice of M >0, let us call the corresponding systemF the(M, ε, L, R)-system inR^d, and denote

T(F) = max{k`_ik |i= 1, . . . , k}.

Our next claim is the following.

Proposition3.4. — Let ε >0. There existδ=δ(ε)>0 andM₀ such that if F is a (M, ε, L, R)-system for a subspaceLofR^d,M >M0 andR>K, then for allx∈LR

with kxklarge enough there exists`∈ F such that for everyv∈BK(`) we have

kx−vk6kxk −δM 4 . Proof. — Let us first assume on the systemF the following:

– F is symmetric, i.e., if`∈ F then−`∈ F, – F ⊂L.

We will also assume thatx∈Landv∈ F.

Our next step is to observe that there exists δ = δ(ε) > 0 such that for any z ∈ S(L) = {x∈ L | kxk = 1} there exists v ∈ F with |hz, vi| >δkvk. Indeed, we can assume that all the `<sub>i</sub> ∈ F are of length one. Denote by S<sup>0</sup> the set of k-tuples {`1, . . . , `k}inS(R^d)which areε-well spread. Since it is a closed condition, the setS⁰ is closed. By compactness of

U ={(z, `1, . . . , `k)|z∈Span(`1, . . . , `k),kzk= 1,(`1, . . . , `k)∈S⁰}

it follows that there exist (`<sup>0</sup><sub>1</sub>, . . . , `⁰_k)∈S⁰,z₀∈Span(`<sup>0</sup><sub>1</sub>, . . . , `⁰_k)withkz0k = 1, and 16i06ksuch that

min

{z,`1,...,`_k}∈U max

16i6k|hz, `ii|= max

16i6k|hz0, `<sup>0</sup><sub>i</sub>i|=|hz0, `⁰_i0i|.

Obviously, the right hand side is positive, since otherwise, we will have that z₀ 6∈

Span(`<sup>0</sup><sub>1</sub>, . . . , `⁰_k). Then we defineδ=|hz0, `⁰_i

0i|.

Letx∈Land let us consider the triangle with the vertices at the origin,xand at v∈ F with⁽²⁾ hx, vi>δkxkkvk. DenoteD=kvk. Notice that D6T(F). We have

kx−vk²=kxk²+D²−2hx, vi.

Assume thatkxksatisfies:

2δkxk −(T(F))²>δkxk,

(2)SinceF is symmetric, suchvexists.

and

kxk>T(F).

Then we have

kxk − kx−vk= 2hx, vi −D²

kxk+kx−vk >δkxkD 3kxk >δM

3 .

For a general (M, ε, L, R)-system F we can find a symmetric (M, ε/2, L, R)-sys- temF⁰ with F⁰ ⊂L, such that for every `<sup>0</sup> ∈ F<sup>0</sup> there exists` ∈ F with k`−`⁰k6 R+K. Takex∈LRwithkxklarge. Then there existsx⁰ ∈Lsuch thatkx−x⁰k6R.

By the previous discussion, there exists δ = δ(ε/2) such that for any x⁰ ∈ L there exists`⁰∈ F⁰ with

kx⁰k − kx⁰−`⁰k> δM 3 .

Take`∈ F such thatk`−`<sup>0</sup>k6R+K. Then for everyv∈BK(`)we have kxk − kx−vk>(kx⁰k − kx⁰−xk)−(kx−x⁰k+kx⁰−`<sup>0</sup>k+k`⁰−`k+k`−vk)

>(kx⁰k − kx⁰−`<sup>0</sup>k)−(2kx−x<sup>0</sup>k+k`⁰−`k+k`−vk)

>δM

3 −3R−2K> δM 4 , where the last transition is correct ifM is large enough.⁽³⁾ 3.1. Proof of Theorem 2.2. — Assume that Λ⊂R^d is an infinite discrete approx- imate subgroup satisfying Λ−Λ ⊂Λ +F for a finite set F. Denote K = diam(F) and L= Span(D(Λ)). Then by Lemma 3.3 there exists R > 0 such thatΛ⊂LR = S

x∈LB_R(x). By the discussion above, there existsε > 0 such that for an arbitrary M >0there exists(M, ε, L, R)-systemF withinΛ. Let us takeM >0so large that the claim of Proposition 3.4 holds true for someδ =δ(ε)>0. LetR⁰ be such that for everyx∈LR withkxk>R⁰ there exists`∈ F with the property that for every v∈B<sub>K</sub>(`)we have:

kx−vk6kxk −δM 4 .

We will show that for every z ∈ L_R we will have B_R⁰(z)∩Λ 6=∅. Assume, on the contrary, that there existsz∈L_R such thatB_R⁰(z)∩Λ =∅. Take minimal R₂> R⁰ such thatBR₂(z)∩Λ6=∅. This means that for everyr < R2we haveBr(z)∩Λ =∅, and that there existsy∈BR₂(z)∩Λ.

Let us denote x=z−y. Then kxk =R₂, and therefore there exists ` ∈ F ⊂Λ such that for every v∈BK(`)we have

kx−vk6kxk − δM

4 <kxk=R₂.

But, since Λ is an approximate subgroup with diam(F) = K, we have that there existsv∈B_K(`)such thaty+v∈Λ. This implies:

kz−(y+v)k< R₂.

(3)Here we use thatδis independent ofK, RandM.

Therefore, there existsr < R2 such that Br(z)∩Λ6=∅. So, we get a contradiction.

Therefore, indeed, for everyx∈LRwe have BR⁰(x)∩Λ6=∅. This finishes the proof

of the theorem.

3.2. Proof of Theorem2.3

“⇒”: IfΛ⊂R^d is an infinite discrete approximate subgroup, then by Theorem 2.2 the setΛis relatively dense around a certain subspaceL⊂R^d. Therefore, there exists R >0 such that

– Λ⊂L+B_R(0_Rd),

– For everyz∈Lwe have Λ∩BR(z)6=∅.

Let L^⊥ ⊂ R^d be the orthogonal complement of L. Let Γ ⊂ L^⊥ be a lattice such that for any x, y ∈ Γ we have kx−yk_L⊥ > 4R. Denote Λ⁰ = Λ + Γ. Obviously, Λ = Λ⁰∩(L+BR(0_Rd)). We claim thatΛ⁰is a Meyer set inR^d, i.e., discrete relatively dense approximate subgroup. Indeed, first notice that

Λ⁰−Λ⁰= (Λ−Λ) + (Γ−Γ)⊂(Λ + Γ) +F = Λ⁰+F,

for some finite set F ∈R^d. Also,Λ⁰ is discrete, since all different translates ofΛ by elements ofΓare far apart. Ifλ1, λ2∈Λ⁰, assume thatλ1=`1+γ1, λ2=`2+γ2 for

`i∈Λ,γi∈Γ, i= 1,2, then

λ₁−λ₂= (`<sub>1</sub>−`₂) + (γ₁−γ₂).

If γ₁ 6= γ₂, then kλ₁ −λ₂k > 2R. And in the case γ₁ = γ₂ we use the uniform discreteness of Λto obtain a uniform bound on kλ1−λ2k, for λ16=λ2. Finally, the relative density of Λ⁰ follows immediately from the relative density ofΛ around the subspace Land the relative density ofΓ insideL^⊥.

“⇐”: LetΛ⁰⊂R^d be a Meyer set. LetR >0be such that for anyx∈R^dwe have B_R/2(x)∩Λ⁰ 6=∅. Take any linear subspaceL⊂R^d. DenoteΛ = Λ⁰∩(L+BR(0_Rd)).

Then Λ is an infinite discrete approximate subgroup. The only non-trivial claim is thatΛis an approximate subgroup. To prove it, we will use Lagarias’ theorem saying that if Λ⁰⁰ is relatively dense inR^d and Λ⁰⁰−Λ⁰⁰is uniformly discrete, then Λ⁰⁰ is an approximate subgroup, i.e., there exists a finite setF ⊂R^dsuch thatΛ⁰⁰−Λ⁰⁰⊂Λ⁰⁰+F. First, we construct such Λ⁰⁰. Take a lattice Γ ⊂ L^⊥ satisfying that for any distinct γ₁, γ₂∈Γwe havekγ₁−γ₂k>4R. Then defineΛ⁰⁰= Λ+Γ. Obviously,Λ⁰⁰is relatively dense inR^d. We also have:

Λ⁰⁰−Λ⁰⁰⊂(Λ⁰−Λ⁰)∩(L+B2R(0_Rd)) + Γ.

This implies thatΛ⁰⁰−Λ⁰⁰ is uniformly discrete, and therefore, by Lagarias theorem, there exists a finite set F⊂R^d withΛ⁰⁰−Λ⁰⁰⊂Λ⁰⁰+F. The latter implies that

Λ−Λ + Γ⊂Λ + Γ +F.

We claim that there exists F⁰ ⊂R^d finite such thatΛ−Λ⊂Λ +F. Indeed, for any

`1, `2∈Λthere exist`3∈Λ, γ ∈Γandf ∈F such that

`1−`2=`3+γ+f.

But the projection ofΛ−Λ−Λ−F ontoL^⊥is at bounded distance from the origin, i.e., there exists R⁰ > 0 such that π_L⊥(Λ−Λ−Λ−F) ⊂ BR⁰(0_Rd)∩L^⊥, where the operatorπ_L⊥ is the orthogonal projection ontoL^⊥. This implies that every such γ ∈ Γ for which there exist `<sub>1</sub>, `₂, `<sub>3</sub> ∈ Λ and f ∈ F with γ = `₁−`<sub>2</sub>−`₃−f is at bounded distance from the origin inL^⊥. But there are only finitely manyγ ∈Γ which lie in the ballBR⁰(0_Rd)∩L^⊥. Denote by F2 the finite set F2 = Γ∩BR⁰(0_Rd), and byF⁰=F+F₂. Then we have

Λ−Λ⊂Λ +F⁰.

3.3. Proof of Theorem 2.4. — It follows immediately from Theorem 2.2 that if Λ⊂Z^d is an infinite approximate group, then there exists a subspace L ⊂ R^d and R >0 such thatΛ⊂L+B_R(0_Rd), and for every`∈Lwe have thatΛ∩B<sub>R</sub>(`)6=∅. Let us call anyΛthat satisfies these constraints with respect to a subspaceLas being relatively dense aroundL.

On the other hand, assume that Λ ⊂ Z^d is relatively dense around a subspace L⊂R^d. We will show that suchΛ is necessarily an approximate subgroup.

Indeed, let us first takeR1>0with the property⁽⁴⁾ that for any pointx∈R^d we have BR₁(x)∩Z^d6=∅. Since, for anyλ∈Λthere exists`∈L such thatλ∈BR(`), we have that for anyλ₁, λ₂∈Λthere existx₁, x₂∈Z^d∩L+B_R1(0)such that

λ_i ∈B_R+R1(x_i), fori= 1,2.

Therefore, there existf₁, f₂∈B_R+R1(0)∩Z^d such that λi=xi+fi, for1 = 1,2.

Also, notice that x1−x2 ∈ L+B2R₁(0). Therefore, there exists λ ∈ Λ such that x1−x2 ∈B3R(λ). Thus, there exists f⁰ ∈B3R(0)∩Z^d such that x1−x2 =λ+f⁰. Finally, let us denoteF =B_5R+2R1(0)∩Z^d (finite set). Then we have

λ₁−λ₂= (x₁+f₁)−(x₂+f₂) = (x₁−x₂) + (f₁−f₂) =λ+ (f₁−f₂+f⁰)∈Λ +F.

This finishes the proof of the Theorem.

3.4. Proof of Proposition2.5. — LetΛ be a discrete approximate subgroup inR^d. By Theorem 2.2 we know that there exist a subspace L and R > 0 such that Λ is relatively dense around L, i.e., Λ ⊂ L+B_R(0_Rd) and for any x ∈ L we have BR(x)∩Λ6=∅. Let us denote byπthe orthogonal projection fromR^d toL. And let ΛL =π(Λ).

By linearity of the map π we get that Λ_L is an approximate subgroup. For

`<sub>1</sub>, `₂ ∈ Λ_L there existλ₁, λ₂ ∈Λ such that `_i =π(λ_i), i= 1,2. Denote by L^⊥ the orthogonal complement toL, i.e., we haveR^d=L⊕L^⊥. Then there existµ1, µ2∈L^⊥ such that

λ_i=`_i+µ_i, fori= 1,2.

(4)We can take anyR1>√ d/2.

ButΛ is an approximate subgroup. Therefore, there exists a finite set F ⊂R^d such thatΛ−Λ⊂Λ +F. This implies that there existλ∈Λ,andf ∈F such that

λ1−λ2=λ+f.

By projecting both sides onLwe obtain:

`1−`2=π(λ) +π(f).

Let us denoteF⁰=π(F)(a finite set). Then we have Λ_L−Λ_L⊂Λ_L+F⁰.

We also have thatΛ_L is relatively dense inL sinceL⊂Λ_L+B_2R(0_Rd).

The set ΛL is discrete. Indeed, assume that it is not discrete. Then there exists (`n) ⊂ ΛL with `n → x ∈ L and `n 6= x for every n. Let (µn) ⊂ L<sup>⊥</sup> such that λn=`n+µn ∈Λ. Since all µn are bounded, then there is a convergent subsequence (µ_nk). Denote its limit byµ∈L^⊥. Then we have

λ_nk =`_nk+µ_nk −→x+µ.

SinceΛis discrete, this implies that the sequenceλ_nk is fixed forklarge enough. This implies that the subsequence`_nkis fixed forklarge enough and we get a contradiction.

All this together, shows that the setΛL ⊂L is a Meyer set. Finally, by the con-

struction we haveΛ⊂ΛL+BR(0_Rd).

4. Discrete approximate subgroups in the Heisenberg group

Assume thatn >1, and Λ⊂H2n+1 is a discrete infinite approximate subgroup.

Denote Λ_V = π_V(Λ). Our fist claim follows from the definition of an approximate group and the linearity of the projection operatorπV

Lemma4.1. — The setΛV is an approximate subgroup in V.

Since the proof of Theorem 2.2 does not use the discreteness of an approximate subgroup inV but only its unboundedness, we derive that there exists a linear sub- spaceL⊂V such that Λ_V is relatively dense aroundL. Our next claim will use the identity (1).

Lemma4.2. — If ω(ΛV,ΛV)6= 0 anddimL>1, then[Λ,Λ]is relatively dense inZ, andΛ_V is discrete inV.

Proof. — By the identity (1), it follows that for any two elementsλ₁= (v, z), λ₂= (u, t), their commutator

(3) [λ1, λ2] = (0, ω(v, u)).

Also, by the assumptions of the lemma, there exist a line L₀ in V, and R >0 such that for every`∈L0 there exists v` ∈ΛV withk`−v`kV 6R. It is also clear from the assumptions that there existsv∈ΛV such thatv6∈L0. Then it follows from the continuity of the symplectic formω that the set

{ω(v, u)|u∈ΛV}

is relatively dense inR. The identity (3) implies that[Λ,Λ]is relatively dense inZ. The only remaining part of the lemma that we have to prove is the discreteness ofΛ_V. SinceΛ is an approximate group inH_2n+1, it follows that there exists a finite setF⁰⊂H (F⁰ =F F F) such that

[Λ,Λ]⊂F⁰Λ.

Thus there exists a relatively dense sequence (tn) ⊂R such that for every ncorre- sponds at least onefnfrom the finite setF⁰⁻¹withfn(0, tn)∈Λ. Assume thatΛV is non-discrete. Then there exists a sequence(v_n, z_n)∈Λwithv_n→v such thatv_n6=v for all n. Then by applying from the left the elementsfn(0, tn) withtn+zn is in a compact set inRwe have the new sequence

fn(vn, tn+zn)⊂F⁰⁻¹FΛ.

But now we achieved that the new sequence is inside a compact set inH_2n+1. Thus, without loss of generality, we assume that the sequence fn(vn, tn+zn) converges.

Since fn’s belong to a finite set F⁰⁻¹, by taking a subsequence, we can assume that fn =f and (vn, tn+zn) converges to(v, t)for some t∈R. Since the element fn is fixed, there exists a finite set F⁰⁰ such that(v_n, t_n+z_n)⊂F⁰⁰Λ. But the set on the right hand side is discrete, while the sequence on the left hand side is not. We get a

contradiction and it finishes the proof of the lemma.

4.1. Proof of Theorem 2.7. — Let Λ be an infinite discrete approximate subgroup in the Heisenberg groupH2n+1. As we already noticed, the projectionΛV ofΛontoV is relatively dense around a subspaceL⊂V. IfL={0}, then by the unboundedness of Λ_V and using the same reasoning as in the proof of Proposition 3.1 we obtain thatΛis relatively dense around the centreZ ofH2n+1. Now assume thatdimL>1.

Then there are two cases:

(1) ω(ΛV,ΛV) = 0, (2) ω(ΛV,ΛV)6= 0.

In the first case, there exists a Lagrangian subspaceL⁰⊂V such thatΛV ⊂L⁰, and ω(L⁰, L⁰) = 0. Then we make use of Schreiber’s theorem with respect to the abelian groupV⁰ =L⁰×Z and conclude that there exists a subspaceL⁰⁰⊂V⁰ such thatΛis relatively dense aroundL⁰⁰. This abelian subgroupL⁰⁰is clearly a connected subgroup ofH2n+1.

In the second case, we invoke Lemma 4.2 and obtain that Λ is relatively dense around the connected subgroupH⁰=LZ, where

LZ ={(v, z)|v∈L, z∈R}.

To prove the last part of the theorem, we notice thatH⁰ around which the subgroupΛ is relatively dense is non-abelian only in the last case, i.e.,H⁰={(v, z)|v∈L, z∈R},

andω(L, L)6= 0. Then by Lemma 4.2 we are done.

4.2. Proof of Proposition 2.10. — If ΛV = πV(Λ) is relatively dense in V, then ω(ΛV,ΛV)6= 0. By Lemma 4.2, we get that[Λ,Λ]is relatively dense inZ. This easily

implies the conclusion of the proposition.

4.3. Proof of Proposition2.11. — LetΛbe as in the statement of the proposition.

Assume that there existsΛ⁰⊂H3 such thatΛ⊂Λ⁰ andΛ⁰ is relatively dense inH3. Then the projectionΛ⁰_V ofΛ⁰ ontoV is non discrete. On other hand, it follows from Lemma 4.2 thatΛ⁰_V is discrete. We get a contradiction.

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Manuscript received 6th February 2018 accepted 9th January 2019

Alexander Fish, School of Mathematics and Statistics F07, University of Sydney NSW 2006, Australia

E-mail :[email protected] Url :http://www.maths.usyd.edu.au/u/afish/