A Weighted Prékopa–Leindler Inequality and Sumsets with Quasicubes

Abstract

AbstractWe 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 \(|A + B + U| \geqslant |A|{ }^{1/2} |B|{ }^{1/2} |U|\). This result is a key ingredient in forthcoming work of the fifth author and Pälvölgyi on the sum-product phenomenon.2000 Mathematics Subject ClassificationPrimary 11B30