On the cardinality of sumsets in torsion-free groups

TL;DR

This paper establishes lower bounds on the size of sumsets in torsion-free groups, showing that large enough subsets force sumsets to grow significantly unless contained in a cyclic subgroup, with explicit bounds on the growth rate.

Contribution

The paper proves a new lower bound on sumset sizes in torsion-free groups, providing explicit bounds on the threshold size for sumset growth and characterizing cases of containment in cyclic subgroups.

Findings

For large enough B, |AB| ≥ |A| + |B| + k unless A is contained in a cyclic subgroup.

Explicit bounds c(k) < c₀k⁶ for general torsion-free groups and c(k) < c₀k³ for groups with the unique product property.

Examples show c(k) is at least quadratic in k.

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 $k$.