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

Abstract

Freiman's 2.4-Theorem states that any set $A \subset \mathbb{Z}_p$ satisfying $|2A| \leq 2.4|A| - 3 $ and $|A| < p/35$ can be covered by an arithmetic progression of length at most $|2A| - |A| + 1$. A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying $|2A| \leq 3|A| - 4$ 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 $|2A| \leq 2.48|A| - 7$ with the more modest density requirement of $|A| < p/10^{10}$.