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arXiv:1009.6140v1 [math.GR] 30 Sep 2010

On the cardinality of sumsets in torsion-free groups

K´ aroly J. B¨ or¨ oczky∗, P´ eter P. P´ alfy†, Oriol Serra

Abstract

Let A,B be finite subsets of a torsion-free group G. We prove that

for every positive integer k there is a c(k) such that if |B| ≥ c(k) then

the inequality |AB| ≥ |A| + |B| + k holds unless a left translate of A

is contained in a cyclic subgroup. We obtain c(k) < c0k6for arbitrary

torsion-free groups, and c(k) < c0k3for groups with the unique product

property, where c0is an absolute constant. We give examples to show

that c(k) is at least quadratic in k.

1 Introduction

Let G be a torsion-free group written multiplicatively, and let | · | denote

the cardinality of a finite set. A basic problem in Additive Combinatorics

is to estimate the cardinality of AB = {ab : a ∈ A,b ∈ B} of two finite sets

A,B ⊂ G in terms of |A| and |B|. A basic notion is a progression with ratio

r ?= 1 and length n, which is a set of the form {a,ar,...,arn−1} where a

and r commute.

Let us review some related results if G is abelian. In this case we have the

simple inequality

|AB| ≥ |A| + |B| − 1,

with equality if and only if A and B are progressions with common ratio.

Following Ruzsa [21], we call the minimal rank of a subgroup whose some

coset contains A the dimension of A. According to Freiman [5], if the di-

mension of A is d, then

|A2| ≥ (d + 1)|A| −

?d + 1

2

?

. (1)

∗Supported by OTKA grants 068398 and 75016, and by the EU Marie Curie FP7 IEF

grant GEOSUMSETS

†Supported by OTKA grant NK72523

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This estimate is optimal.

contained in some coset of a cyclic group. Actually, even a progression of

length 2|A|−3 contains A according to the (3k−4)-theorem of Freiman [5].

More precise structural information on A is available if |A2| = 2|A| + n for

0 ≤ n ≤ |A|−4 by Freiman [7], for example, A is contained in a progression

of length |A| + n + 1.

It follows that if |A2| ≤ 3|A| − 4, then A is

The inequality (1) was generalized to a pair of sets by Ruzsa [20] who proved

that if |A| ≥ |B|, and the dimension of AB is d, then

|AB| ≥ |A| + d|B| −

?d + 1

2

?

. (2)

By requiring additionally that the smaller set B is d–dimensional, Gardner

and Gronchi [8] proved a discrete version of the Brunn–Minkowski inequality

which shows that

|AB| ≥ |A| + (d − 1)|B| + (|A| − d)(d−1)/d(|B| − d)1/d−

?d

2

?

. (3)

Additional lower bounds with stronger geometric requirements on the sets

A and B have been also obtained by Matolcsi and Ruzsa [16] and Green and

Tao [9].

In the non-abelian case the situation is much less understood. Kempermann

[15] implies in the case of any torsion-free group G that

|AB| ≥ |A| + |B| − 1. (4)

Brailovsky and Freiman [1] characterized the extremal sets in the inequality

(4) by showing that, if min{|A|,|B|} ≥ 2 then, up to appropriate left and

right translations, both A and B are progressions with common ratio. In

particular, A and B lie in a left and a right coset, respectively, of a cyclic

subgroup.

The analogy with the abelian case was extended in Hamidoune, Llad´ o and

Serra [14] to the inequality

|AB| ≥ |A| + |B| + 1, (5)

if |B| ≥ 4, and A is not contained in some left coset of a cyclic subgroup.

These known facts are connected with the following conjecture of Freiman

(personal communication), extending the (3k − 4)-theorem above.

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Conjecture 1 Let A be a finite subset of a torsion-free group with |A| ≥ 4.

If

|A2| ≤ 3|A| − 4

then A is covered by a progression of length at most 2|A| − 3.

By using the so–called isoperimetric method, see Hamidoune [11], [12] or

[13], we obtain the following results:

Theorem 2 For any integer k ≥ 1 there exists a c(k) such that the following

holds. If G is a torsio-free group, A ⊂ G is not contained in a left coset of

any cyclic subgroup, B ⊂ G has more than c(k) elements, then

|AB| > |A| + |B| + k.

Remark Our current methods yield c(k) ≤ 32(k + 3)6.

Note that, in Theorem 2, the assumption on A not being contained in a left

coset of a cyclic group is crucial. For example, if A is a progression of length

at most k + 2 with ratio r ?= 1, and B is the union of two r-progressions of

arbitrary length, then |AB| ≤ |A| + |B| + k.

The value of the lower bound c(k) can be improved for unique product

groups. Recall that a group G has the unique product property if, for ev-

ery pair of finite sets A,B ⊂ G, there is an element g ∈ AB which can be

uniquely expressed as a product of an element of A and an element of B.

In this case, G is torsion-free. We note that every right linearly orderable

group has the unique product property, and any residually finite word hy-

perbolic group has a finite index unique product subgroup, according to T.

Delzant [4]. On the other hand, it was first shown by Rips and Segev [18]

that not all torsion-free groups have the unique product property, and S.D.

Promislow [17] even provided an explicit construction for such an example.

In addition, unique product groups are discussed in A. Strojnowski [22],

S.M. Hair [10], and W. Carter [3]. For unique product groups, the bound

on c(k) in Theorem 2 can be reduced to a cubic polynomial on k; namely,

Lemma 12 yields that

c(k) ≤ 4(2k + 3)3if G is a unique product group.(6)

We note that it can be deduced with the help of (3) that in abelian torsion-

free groups, the optimal order of c(k) is quadratic. The c(k) in Theorem 2

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is at least of quadratic order also for non-abelian unique product groups, as

the following example shows.

We consider the Klein bottle group G0 = ?u,v|u−1vu = v−1?, and hence

vu = uv−1and v−1u = uv. Since ?v? is a normal subgroup with factor

isomorphic to Z, G0is a non-abelian unique product group. Let A = {1,u,v}

and B = {uivj: i,j = 0,1,...,m − 1} for m ≥ 1. Then |B| = m2and

AB = ∪i,j=0,1,...,m−1{uivj,ui+1vj,uivj+(−1)i}, thus

|AB| = m2+ 2m = |A| + |B| + 2|B|

1

2− 3.

In line with Conjecture 1, we conjecture that, if A = B, then the lower

bound on |B| in Theorem 2 can be replaced by a bound linear in k. We

construct an example in the group G0above to indicate, what to expect in

Theorem 2 in this case.

For m ≥ 1, let A = P ∪ vuQ where P = {ui: i = 0,1,...,2m} and Q =

{u2i: i = 0,1,...,m − 1}, and hence |A| = 3m + 1. Since v commutes with

u2, we have (vuQ)(vuQ) = (vuv)uQ2= u2Q2⊂ P2. Moreover, denoting by

P0and P1the set of even and odd powers of u in P, respectively, we have

PvuQ = vuP0Q ∪ v−1uP1Q ⊂ vuQP ∪ v−1uP1Q. It follows that

|A2| = |P2| + |QP| + |P1Q| = 10m − 1 =10

3|A| −13

3.

We note that the above example seems to match a conjecture of Freiman

[6], which would yield that A is the union of two progressions provided that

|A2| <10

3|A| − 5.

In the direction of Conjecture 1 for a torsion-free group G, our results yield

the following.

Corollary 3 If A is a subset of a torsion-free group with |A| ≥ 66, and

|A2| = 2|A| + n for 0 ≤ n ≤ 2−5/6|A|1/6− 3, then A is contained in a

progression of length |A| + n + 1.

Remark

0 ≤ n ≤ 2−5/3|A|1/3−3

In unique product groups, the conditions are |A| ≥ 63and

2.

If A is a finite subset of a torsion-free group G, then Corollary 3 provides

strong structural information when |A2| is very close to 2|A|. This has been

made possible in part by the known structural properties in abelian groups.

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Now if G is abelian and |A2| < K|A| for some K > 3 then still strong struc-

tural properties have been established by Freiman [5] using multidimensional

progressions, see the monograph of Tao and Vu [24] or the survey by Ruzsa

[21] for recent developments. But if G is any torsion-free group and K ≥10

then A may not be contained in an abelian subgroup. Actually, it is still

not completely understood, what to expect, in spite the results about some

specific groups (see Breuillard and Green [2], or Tao’s blog [23]).

3,

2 Atoms and fragments

For this section, we fix a torsion-free group G.

For n ≥ 1 and a finite non-empty set C ⊂ G, the n-th isoperimetric number

of C is defined to be

κn(C) = min{|XC| − |X| : X ⊂ G and |X| ≥ n}.

A finite set V ⊂ G is an n-fragment for C, if |V | ≥ n and |V C|−|V | = κn(C).

In addition an n-fragment of minimal cardinality is an n-atom for C.

Naturally, if U is an n-atom for C, then xU is also an n-atom for Cy for

any x,y ∈ G. In what follows, we present simple statements about atoms.

For the sake of completeness, we verify even the known ones, except for

the following crucial property of atoms, due to Hamidoune [11]: If U is an

n–atom and F is an n–fragment for a finite nonempty subset C ⊂ G, then

either U ⊂ F or |U ∩ F| ≤ n − 1.

This property has the following useful consequence.

Corollary 4 For a torsion-free group G and n ≥ 1, if U is an n-atom for

C ⊂ G and g ∈ G\1, then |U ∩ gU| ≤ n − 1.

For right translations we have a weaker result.

Lemma 5 For a torsion-free group G and n ≥ 2, if U is an n-atom for

C ⊂ G and g ∈ G\1, then

|U ∩ Ug| ≤n − 2

n − 1|U| +

1

n − 1≤n − 1

n

|U|.

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