On Minkowski product size: The Vosper's property

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 $|S^{-1}S|,|S{S}^{-1}|<2|S|,|G^{\prime }|-1,$ we show that there exist an element $a\in S,$ and a non-null subgroup $H$ of $G^{\prime }$ such that either $S^{-1}HS=S^{-1}S\cup a^{-1}Ha$ or $SH{S}^{-1}=S{S}^{-1}\cup aH{a}^{-1},$ where $G^{\prime }$ is the subgroup generated by $S^{-1}S.$

References in corpus (1)

• A Structure Theory for Small Sum Subsets

Cited by in corpus (2)

• The Structure of Critical Product Sets
• A Structure Theorem for Small Sumsets in Nonabelian Groups