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

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 $\mathrm{\mid }A\mathrm{\mid }<0.0045p$, (i) if $\mathrm{\mid }A+A\mathrm{\mid }<2.59\mathrm{\mid }A\mathrm{\mid }-3$ and $\mathrm{\mid }A\mathrm{\mid }>100$, then $A$ is contained in an arithmetic progression of size $\mathrm{\mid }A+A\mathrm{\mid }-\mathrm{\mid }A\mathrm{\mid }+1$, and (ii) if $\mathrm{\mid }A-A\mathrm{\mid }<2.6\mathrm{\mid }A\mathrm{\mid }-3$, then $A$ is contained in an arithmetic progression of size $\mathrm{\mid }A-A\mathrm{\mid }-\mathrm{\mid }A\mathrm{\mid }+1$. The improvement comes from using the properties of higher energies.