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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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Remark In particular, if n = 2, then |U ∩ Ug| ≤ 1.
Proof:
Let us partition U into the maximal left g-progressions U1,...,Um,
where Ui= {hi,hig,...,higαi}, i = 1,...,m. In particular, Ui∩Ujg = ∅ for
i ?= j. We may assume that |U1| ≥ |Ui|, i = 2,...,m and that h1= 1.
It follows by Corollary 4 that
|U1∩ gU1| ≤ n − 1,
thus |U1| ≤ n. In addition, for i ≥ 2, we have
|U1∩ h−1
iUi| ≤ n − 1,
thus |Ui| ≤ n − 1. Therefore |U| ≤ m(n − 1) + 1 and
|U ∩ Ug| = |U| − m ≤n − 2
n − 1|U| +
1
n − 1≤n − 1
n
|U|,
as claimed. Q.E.D.
The minimality of the cardinality of atoms directly yields (see [11] or [14])
Lemma 6 If U is an n-atom for C ⊂ G, n ≥ 1, in a torsion-free group
G, and |U| > n, then any element in UC can be represented in at least two
ways as a product of an element of U and an element of C.
We deduce two rough, but useful estimates about 2-atoms which can be
found in [11] as well.
Lemma 7 If U is a 2-atom for C ⊂ G, |C| ≥ 3, in a torsion-free group G,
then |U| ≤ |C| − 1.
Proof:
According to Lemma 6, for any u ∈ U, there are vu∈ U and cu∈ C\1 such
that u = vucu. If cu= cw for u ?= w ∈ U, then {vw,w} ⊂ U ∩ vwv−1
contradicting Corollary 4. Therefore u ?→ cuis an injective map from U into
C\1. Q.E.D.
We may assume that |U| > 2 and 1 ∈ C, and hence U ⊂ UC.
uU,
For any non-empty C ⊂ G, let C−1= {g−1: g ∈ C}.
Lemma 8 If U is a 2-atom for C ⊂ G, |C| ≥ 3, in a torsion-free group G,
and |UC| ≤ |U| + |C| + k, then |U| ≤ k + 3.
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Proof:
Let V be a 2-atom for U−1with 1 ∈ V , thus
|V U−1| − |V | − |U−1| ≤ |C−1U−1| − |C−1| − |U−1| = |UC| − |U| − |C| ≤ k.
If V = {1,g} with g ?= 1, then Lemma 5 yields
2|U| − 1 ≤ |UV−1| = |V U−1| ≤ k + 2 + |U|,
which in turn implies |U| ≤ k + 3. If |V | ≥ 3, then Lemma 5 and Lemma 7
yield
|U|+(|U|−1)+(|U|−2) ≤ |UV−1| = |V U−1| ≤ k+|V |+|U| ≤ k+2|U|−1,
which in turn implies |U| ≤ k + 2. Q.E.D.
Now we extend Lemma 8 to n-atoms, which extension is the only novel
result of this section.
Proposition 9 If U is an n-atom for C ⊂ G, |C| ≥ 3 and n ≥ 3, in a
torsion-free group G, and |UC| ≤ |U| + |C| + k, then |U| ≤ n(2k + 3).
Proof:
Let V be a 2-atom for U−1, hence
|V U−1| − |V | − |U−1| ≤ |C−1U−1| − |C−1| − |U−1| ≤ k.
It follows by Lemma 8 that |V | ≤ k + 3. Moreover, by Lemma 5, we have
|UV−1| ≥ 2|U| −n−1
n|U|. Hence,
n+1
n|U| ≤ |UV−1| = |V U−1| ≤ |U| + |V | + k ≤ |U| + 2k + 3,
thus |U| ≤ n(2k + 3). Q.E.D.
All these statements about atoms would readily follow from the following
conjecture of Y.O. Hamidoune [13].
Conjecture 10 Any n-atom in a torsion-free group has cardinality n.
We recall that a group G has the unique product property if for any finite
non-empty sets A,B ⊂ G, there is a g ∈ AB that can be represented in
a unique way in the form ab with a ∈ A and b ∈ B. In this case G is
torsion-free.It follows by Lemma 6 that unique product groups satisfy
Conjecture 10.
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3 Small product sets
The proof of Theorem 2 together with an estimate of c(k) will follow from
the following Lemma and the estimations on the size of atoms in the previous
section.
Lemma 11 Let G be a torsion-free group. Suppose that A ⊂ G with |A| = 3
is not contained in a left coset of any cyclic subgroup of G. For d ≥ 3 and
any finite set B ⊂ G of cardinality greater than 4d3, we have
|AB| > |B| + d.
Proof: We suppose that |AB| ≤ |B|+d, and seek a contradiction. We may
assume that A = {1,u,v}, where ?u,v? is not cyclic.
For g ∈ G, we write Bg= B \ g−1B = {x ∈ B | gx / ∈ B}. Since |B ∪ uB| ≤
|B| + d, we see that |Bu| ≤ d. Similarly, |Bu−1| ≤ d, as |u−1B ∪ B| =
|B ∪ uB| ≤ |B| + d, and, of course, |Bv|,|Bv−1| ≤ d also hold. Since B is
finite, for any x ∈ B the coset ?u?x must contain an element of Bu, hence
the elements of B belong to at most d cosets of ?u?, and similarly for ?v?.
Therefore there exists an x0∈ B such that
|B ∩ ?u?x0∩ ?v?x0| ≥ |B|/d2> 4d.
In order to simplify notation, by replacing B with Bx−1
without loss of generality that x0= 1. Let Z = ?u? ∩ ?v?, and B0= B ∩ Z.
We have |B0| > 4d. Elements of Z are powers of both u and v, hence Z is
contained in the center of H = ?u,v?. As Z ?= {1} and A does not generate
a cyclic group, we deduce that u and v do not commute.
0, we may assume
We are going to show that BZ ⊇ H. Take an element g ∈ H, and let
us choose a word of shortest length anan−1···a2a1, where each ai is one
of u, u−1, v, v−1, in the coset gZ. Then the cosets Z, a1Z, a2a1Z, ...,
anan−1···a1Z are pairwise disjoint.
quence Sx= {aiai−1···a1x}i=0,1,...,n, which sequences are pairwise disjoint
as x runs through B0. If anan−1···a1x ?∈ B, then there is a smallest
i ∈ {1,...,n} such that ai···a1x ?∈ B. It follows that Sxhas an element in
Bai⊂ Bu∪ Bu−1 ∪ Bv∪ Bv−1, namely, x if i = 1, and ai−1···a1x if i ≥ 2.
Since |B0| > 4d ≥ |Bu| + |Bu−1| + |Bv| + |Bv−1|, and Sx∩ Sy= ∅ for x ?= y
in B0, there exists an x ∈ B0such that anan−1···a1x ∈ B. We conclude
that g ∈ anan−1···a1xZ ⊂ BZ.
To any x ∈ B0, we assign the se-
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Now the index of the central subgroup Z in H is finite (bounded by |B|), so
the center has finite index in H. According to a classical theorem of Schur
(see, e.g., Robinson [19, Theorem 10.1.4]), this implies that the commutator
subgroup of H is finite. If the commutator subgroup is 1, then H is abelian,
and if the commutator subgroup is non-trivial, then we have some torsion
elements. In any case, we have contradicted the assumptions on A and G,
and hence proved the lemma. Q.E.D.
Proof of Theorem 2: Without loss of generality we may assume that 1 ∈ A.
Then ?A? is not cyclic by our assumption. Let B ⊂ G be a finite set with
|B| > 32(k + 3)6. If B is contained in some right coset of a cyclic subgroup
H, then A intersects at least two left cosets of H. Let A1be one of these
intersections. Then using (4) we get
|AB| = |A1B| + |(A\A1)B| ≥ |A| + 2|B| − 2 > |A| + |B| + k.
Therefore we may assume that B is not contained in a right coset of any
cyclic subgroup.
If |A| ≤ k + 3, then let A0 = A, and if |A| > k + 3, then let A0 be a
(k + 3)–atom for B with 1 ∈ A0. By definition, |AB| ≥ |A| + |A0B| − |A0|.
Proposition 9 gives that either |A0B|−|A0| > |B|+k, or |A0| ≤ (k+3)(2k+
3). If ?A0? is not cyclic, then choose u,v ∈ A0\ 1 such that ?u,v? is not
cyclic. For? A0= {1,u,v} ⊂ A0, we have, by Lemma 11, that
|AB| ≥ |A|+|A0B|−|A0| ≥ |A|+|? A0B|−|A0| > |A|+(|B|+2(k+3)2)−|A0| > |A|+|B|+k.
Finally if ?A0? is cyclic, then A0?= A, so |A0| ≥ k + 3, and B intersects at
least two right cosets of ?A0?. Let B1be one of these intersections. We have
by (4)
|AB| ≥ |A|+|A0B|−|A0| = |A|+|A0B1|+|A0(B\B1)|−|A0| ≥ |A|+|B|+|A0|−2 > |A|+|B|+k,
completing the argument. Q.E.D.
If G is a unique product group, then the argument above, just using Con-
jecture 10 in place of Proposition 9, leads to
Lemma 12 Let G be a unique product group, A,B ⊂ G finite subsets, and
k ≥ 1. Suppose that A is not contained in a left coset of any cyclic subgroup,
and |B| > 4(2k + 3)3, then
|AB| > |A| + |B| + k.
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Acknowledgement: We are grateful for the help of Mikl´ os Ab´ ert, Warren
Dicks, G´ abor Elek and Imre Ruzsa in the preparation of this manuscript.
We particularly thank Yahya O. Hamidoune for fruitful discussions on the
problem addressed in this paper, and Peter A. Linnell for providing in depth
information on unique product groups.
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K´ aroly J. B¨ or¨ oczky, carlos@renyi.hu
Alfr´ ed R´ enyi Institute of Mathematics, and
Universitat Polit` ecnica de Catalunya, Barcelona Tech, and
Department of Geometry, Roland E¨ otv¨ os University
P´ eter P. P´ alfy, ppp@renyi.hu
Alfr´ ed R´ enyi Institute of Mathematics
Oriol Serra, oserra@ma4.upc.edu
Universitat Polit` ecnica de Catalunya, Barcelona Tech
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