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 $|2A|\le 2.4|A|-3$ 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 $|2A|\le 3|A|-4$ 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 $|2A|\le 2.48|A|-7$ with the more modest density requirement of $|A|<p/10^{10}$.

13 pages, 1 figure