On Ruzsa's discrete Brunn-Minkowski conjecture

Abstract

We prove a conjecture by Ruzsa from 2006 on a discrete version of the Brunn-Minkowski inequality, stating that for any $A,B\subset {\mathbb{Z}}^k$ and $\epsilon >0$ with $B$ not contained in $n_{k,\epsilon }$ parallel hyperplanes we have $|A+B{|}^{1/k}\ge |A{|}^{1/k}+\left(1-\epsilon \right)|B{|}^{1/k}$.

6 pages