A STEP BEYOND KNESER'S THEOREM FOR ABELIAN FINITE GROUPS

Abstract

A precise description of a subset $\mathcal{A}$ of $\mathbb{Z} / n \mathbb{Z}$ satisfying$$ | \mathcal{A} + \mathcal{A} | \leq 2.04 | \mathcal{A} | $$is given. Basically, there exists a subgroup $\mathcal{H}$ of $\mathbb{Z} / n \mathbb{Z}$ such that $\mathcal{A}$ is included in an arithmetic progression of $\ell$ cosets modulo $\mathcal{H}$ and$$(\ell - 1) | \mathcal{H} | \leq | \mathcal{A} + \mathcal{A} | - | \mathcal{A} |.$$ 2000 Mathematical Subject Classification: 11B50, 11B83, 20E34.