On Ruzsa's discrete Brunn-Minkowski conjecture

TL;DR

This paper proves Ruzsa's 2006 conjecture on a discrete version of the Brunn-Minkowski inequality for subsets of integer lattices, establishing a lower bound on the sumset size relative to the original sets.

Contribution

The paper provides a proof of Ruzsa's conjecture, offering a new discrete inequality related to the Brunn-Minkowski principle in lattice settings.

Findings

Established a lower bound for sumsets in integer lattices.

Extended the Brunn-Minkowski inequality to discrete settings.

Validated Ruzsa's conjecture for sets not contained in few hyperplanes.

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 $|A+B|^{1/k}\geq |A|^{1/k}+\left(1-\epsilon\right)|B|^{1/k}$.