A Weighted Prékopa-Leindler inequality and sumsets with quasicubes

Abstract

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prékopa-Leindler inequality. This is then applied to show that if $A,B\subseteq {\mathbb{Z}}^d$ are finite sets and $U$ is a subset of a "quasicube" then $|A+B+U|\ge |A{|}^{1/2}|B{|}^{1/2}|U|$. This result is a key ingredient in forthcoming work of the fifth author and Pälvölgyi on the sum-product phenomenon.

5 pages