# On the size of subsets of $\mathbb{F}_p^{n}$ without $p$ distinct elements summing to zero

## TL;DR

This paper establishes a new upper bound on the size of subsets of _p^n that lack p distinct elements summing to zero, improving previous bounds by applying the polynomial method and multi-colored sum-free theorems.

## Contribution

The paper introduces a significantly improved upper bound for subsets avoiding p-sums in _p^n using advanced polynomial techniques and sum-free theorems.

## Key findings

- New upper bound of C_p*(2p)^n for subset size
- Bound of p^{(1/2)(1+o(1))n} as p,n grow large
- Improvement over previous bounds of p^{(1-o(1))n}

## Abstract

Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds.