Towards $3n-4$ in groups of prime order

We show that if $A$ is a subset of a group of prime order $p$ such that $\mathrm{\mid }2A\mathrm{\mid }<2.7652\mathrm{\mid }A\mathrm{\mid }$ and $\mathrm{\mid }A\mathrm{\mid }<1.25\cdot {10}^{-6}p$, then $A$ is contained in an arithmetic progression with at most $\mathrm{\mid }2A\mathrm{\mid }-\mathrm{\mid }A\mathrm{\mid }+1$ terms, and $2A$ contains an arithmetic progression with the same difference and at least $2\mathrm{\mid }A\mathrm{\mid }-1$ terms. This improves a number of previously known results.