On the Erd\H{o}s--Ginzburg--Ziv Problem in large dimension

Abstract

The Erd\H{o}s--Ginzburg--Ziv Problem is a classical extremal problem in discrete geometry. Given $m$ and $n$, the problem asks about the smallest number $s$ such that among any $s$ points in the integer lattice $\mathbb{Z}^n$ one can find $m$ points whose centroid is again a lattice point. Despite of a lot of attention over the last 50 years, this problem is far from well-understood. For fixed dimension $n$, Alon and Dubiner proved that the answer grows linearly with $m$. In this paper, we focus on the opposite case, where the number $m$ is fixed and the dimension $n$ is large. We drastically improve the previous upper bounds in this regime, showing that for every $\varepsilon>0$ the answer is at most $D_{\varepsilon,m}\cdot (C_{\varepsilon}m^{\varepsilon})^n$ for all $m$ and $n$. Our proof combines (a consequence of) the slice rank polynomial method with a higher-uniformity version of the Balog--Szemer\'{e}di--Gowers Theorem due to Borenstein and Croot.