Small doubling in cyclic groups

TL;DR

This paper characterizes subsets of finite cyclic groups with small doubling, showing they are densely contained in coset progressions, thus advancing understanding of additive structure in cyclic groups.

Contribution

It provides a comprehensive description of sets with small doubling in cyclic groups, improving previous results by identifying their dense containment in coset progressions.

Findings

Sets with small doubling are densely contained in coset progressions.

The result generalizes and improves earlier theorems by Deshouillers-Freiman and Balasubramanian-Pandey.

The characterization applies to finite cyclic groups, enhancing structural understanding.

Abstract

We give a comprehensive description of the sets $A$ in finite cyclic groups such that $|2A|<\frac94|A|$; namely, we show that any set with this property is densely contained in a (one-dimensional) coset progression. This improves earlier results of Deshouillers-Freiman and Balasubramanian-Pandey.