Small doubling in prime-order groups: from $2.4$ to $2.6$

TL;DR

This paper improves bounds on the structure of small doubling sets in prime-order groups, showing they are contained in arithmetic progressions under new, tighter conditions by leveraging higher energy properties.

Contribution

It provides sharper bounds for small doubling sets in prime groups, extending previous results with novel use of higher energy techniques.

Findings

Sets with small sumsets are contained in arithmetic progressions.

Improved bounds from 2.4 to 2.6 in doubling constants.

Results apply to subsets of finite prime fields with size constraints.

Abstract

Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset $A\subseteq\mathbb F_p$ with $p$ prime and $|A|<0.0045p$, (i) if $|A+A|<2.59|A|-3$ and $|A|>100$, then $A$ is contained in an arithmetic progression of size $|A+A|-|A|+1$, and (ii) if $|A-A|<2.6|A|-3$, then $A$ is contained in an arithmetic progression of size $|A-A|-|A|+1$. The improvement comes from using the properties of higher energies.