On the cardinality of sumsets in torsion-free groups

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) < c 0k 6 for arbitrary torsion-free groups, and c(k) < c 0k 3 for groups with the unique product property, where c0 is an absolute constant. We give examples to show that c(k) is at least quadratic in k.

A structure theorem for small sumsets in nonabelian groups

“…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].…”

Cardinality of product sets in torsion-free groups and applications in group algebras

“…Also, in [21], it is proved that if B is not contained in the left coset of a cyclic subgroup, and |C| ≥ 32(3 + k) 6 , for k ≥ 1, then…”

Sums of Dilates in Ordered Groups

“…There is, for instance, a conjecture by Freiman stating that the inequality |2X| < 3|X| − 4 for a subset in a torsion-free group implies that the subgroup generated by X is cyclic (and then the structure of X can be recovered by Freiman's 3k − 4 theorem). To our knowledge, the best general inequality for torsion-free groups can be found in [3], where the main result implies that, for any constant k one has |2X| ≥ 2|X| + k provided that |X| is large enough and not contained in a coset of a cyclic subgroup.…”

“…This may indicate that small sets are precisely not the ones where good estimates will be found. However, an explicit example is given in [3] where…”