Sharp bound for the Erd\H{o}s-Straus non-averaging set problem

TL;DR

This paper establishes a sharp upper bound on the size of the largest non-averaging subset of integers up to n, solving the Erd ext{"o}s-Straus problem and extending results to higher dimensions.

Contribution

It provides the first tight bound on the maximum size of non-averaging sets, using advanced structural theorems and geometric analysis.

Findings

Largest non-averaging set size is approximately n^{1/4}

Determines maximum size of non-averaging sets in d-dimensional boxes

Introduces new structural tools for analyzing subset sums

Abstract

A set of integers $A$ is non-averaging if there is no element $a$ in $A$ which can be written as an average of a subset of $A$ not containing $a$. We show that the largest non-averaging subset of $\{1, \ldots, n\}$ has size $n^{1/4+o(1)}$, thus solving the Erd\H{o}s-Straus problem. We also determine the largest size of a non-averaging set in a $d$-dimensional box for any fixed $d$. Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.