On the cardinality of sumsets in torsion-free groups

Bulletin of the London Mathematical Society (Impact Factor: 0.63). 09/2010; DOI: 10.1112/blms/bds032
Source: arXiv

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 $|B|\ge c(k)$ 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 $k$.

  • Source
    [Show abstract] [Hide abstract]
    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.
    European Journal of Combinatorics 03/2012; · 0.66 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    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.
    Journal of Group Theory 01/2013; 3(3). · 0.49 Impact Factor

Full-text (2 Sources)

Available from