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

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.