The sumset phenomenon

Abstract

Answering a problem posed by Keisler and Leth, we prove a theorem in non–standard analysis to reveal a phenomenon about sumsets, which says that if two sets $A$ and $B$ are large in terms of “measure”, then the sum $A+B$ is not small in terms of “order–topology”. The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2–4. One of these is a new result in additive number theory; it says that if two sets $A$ and $B$ of non–negative integers have positive upper or upper Banach density, then $A+B$ is piecewise syndetic.