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

Abstract

Let us fix a prime p. The Erdős-Ginzburg-Ziv problem asks for the minimum integer s such that any collection of s points in the lattice ℤⁿ contains p points whose centroid is also a lattice point in ℤⁿ. 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 ≥ 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)·(1+o(1))n}, whereas all previously known upper bounds were of the form p^(1−o(1))n (with pⁿ 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.