Sharp Bound for the Erdős–Straus Non-averaging Set Problem

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 , … , n } has size n 1 / 4 + o ( 1 ) , thus solving the Erdős–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.