On Minkowski product size: The Vosper's property

TL;DR

This paper characterizes Vosper's subsets in groups, showing their structure under certain conditions and revealing their relation to subgroup cosets, advancing the understanding of sumset properties in group theory.

Contribution

The work provides a structural description of Vosper's subsets assuming they are not progressions and satisfy size constraints, identifying their relation to subgroup cosets.

Findings

Vosper's subsets are structurally linked to subgroup cosets.

Under certain size conditions, Vosper's subsets can be described via specific subgroup relations.

The results extend the understanding of sumset properties in non-abelian groups.

Abstract

A subset $S$ of a group $G$ is said to be a Vosper's subset if $|A\cup AS|\ge \min (|G|-1,|A|+|S|),$ for any subset $A$ of $G$ with $|A|\ge 2.$ In the present work, we describe Vosper's subsets. Assuming that $S$ is not a progression and that $|S^{-1} S|, |S S^{-1}| <2 |S|,|G'|-1,$ we show that there exist an element $a\in S,$ and a non-null subgroup $H$ of $G'$ such that either $S^{-1}HS =S^{-1}S \cup a^{-1}Ha$ or $SHS^{-1} =SS^{-1}\cup aHa^{-1},$ where $G'$ is the subgroup generated by $S^{-1}S.$