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

Abstract

Freiman’s 2.4-Theorem states that any set A¿Zp satisfying |2A|=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|=3|A|-4 as long as the rather strong density requirement |A|<p/10215 is satisfied. We present a version of this statement that allows for sets satisfying |2A|=2.48|A|-7 with the more modest density requirement of |A|<p/1010.