A step beyond Freiman's theorem for set addition modulo a prime

Freiman’s 2.4-Theorem states that any set $A\subset {\mathbb{Z}}_p$ satisfying $\mathrm{\mid }2A\mathrm{\mid }\le 2.4\mathrm{\mid }A\mathrm{\mid }-3$ and $\mathrm{\mid }A\mathrm{\mid }<p\mathrm{/}35$ can be covered by an arithmetic progression of length at most $\mathrm{\mid }2A\mathrm{\mid }-\mathrm{\mid }A\mathrm{\mid }+1$. A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying $\mathrm{\mid }2A\mathrm{\mid }\le 3\mathrm{\mid }A\mathrm{\mid }-4$ as long as the rather strong density requirement $|A| < p/10^{215}$ is satisfied. We present a version of this statement that allows for sets satisfying $\mathrm{\mid }2A\mathrm{\mid }\le 2.48\mathrm{\mid }A\mathrm{\mid }-7$ with the more modest density requirement of $\mathrm{\mid }A\mathrm{\mid }<p\mathrm{/}{10}^{10}$.