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On the cardinality of sumsets in torsion-free groups (1009.6140v1)

Published 30 Sep 2010 in math.GR and math.CO

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 $k$.

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