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 $|B|\ge c(k)$ then the inequality $|AB|\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in a cyclic subgroup. We obtain $c(k)<c_0{k}^6$ for arbitrary torsion-free groups, and $c(k)<c_0{k}^3$ 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 $k$.

References in corpus (3)

• The structure of approximate groups
• Approximate groups, I: the torsion-free nilpotent case
• A nilpotent Freiman dimension lemma

Cited by in corpus (3)

• Cardinality of product sets in torsion-free groups and applications in group algebras
• A Structure Theorem for Small Sumsets in Nonabelian Groups
• The sum-product problem for small sets