On the cardinality of sumsets in torsion-free groups

Bulletin of the London Mathematical Society (Impact Factor: 0.7). 09/2010; 44(5). DOI: 10.1112/blms/bds032
Source: arXiv


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 $|B|\ge c(k)$ then the inequality $|AB|\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in a cyclic subgroup. We obtain $c(k)<c_0k^{6}$ for arbitrary torsion-free groups, and $c(k)<c_0k^{3}$ for groups with the unique product property, where $c_0$ is an absolute constant. We give examples to show that $c(k)$ is at least quadratic in $k$.

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    • "A group is G is said to satisfy the unique product property if given any two non-empty finite sets X, Y ⊂ G then at least one element, say z in the product set XY = {xy | x ∈ X and y ∈ Y } can be written uniquely as a product, z = xy where x ∈ X and y ∈ Y . Finite product sets within these groups are studied in [1]. Many familiar groups satisfy this property, for example, orderable groups [8], diffuse groups [2] and locally indicable groups [4]. "
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    ABSTRACT: We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.
    Preview · Article · Jan 2013 · Journal of Group Theory
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    • "The paper is organised as follows: in the Section 2 we introduce the isoperimetric tools that we shall need in the sequel. We then proceed in Section 3 to obtain a general upper bound for the cardinality of nonperiodic 2–atoms, thus extending an analogous result obtained by Hamidoune [6] for abelian groups, for normal sets in simple groups by Arad and Muzychuk [1] and in [2] for torsion-free groups. Section 4 contains a somewhat shortened account of Hamidoune's result on 2-atoms obtained in [7]. "
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    ABSTRACT: Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the nonabelian case classical results for Abelian groups. When we remove the hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the above characterization whose structure is described precisely.
    Preview · Article · Mar 2012 · European Journal of Combinatorics
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    • "or again a whole class. We just mention two recent results in this regard: the first, which is Theorem 2 in [6], covers torsion free groups G. It states that there exists a polynomial p(n) = 32(n + 3) 6 such that if A is a finite subset of any torsion free group G which is not contained in a left coset of a cyclic subgroup and |B| > p(n) then |AB| > |A| + |B| + n. "
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    ABSTRACT: Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.
    Preview · Article · Jun 2011 · Algebra Colloquium
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