Open Collections

On the size of subsets of $F_p^n$ without $p$ distinct elements summing to zero

Banff International Research Station for Mathematical Innovation and Discovery

Let us fix a prime $p$. The Erdos-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $Z^n$ contains $p$ points whose centroid is also a lattice point in $Z^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $F_p^n$ without $p$ distinct elements summing to zero. In this talk, we discuss a new upper bound for this problem for any fixed prime $p \geq 5$ and large $n$. Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. However, these tools cannot be applied directly in our setting, and there has been work by several authors to apply these tools to the problem discussed here. Using some key new ideas, we significantly improve the previous bounds.

Mathematics; Combinatorics; Probability theory and stochastic processes

video/mp4

Author affiliation: Stanford University

2020-03-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73633

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/