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

A set of integers A A is non-averaging if there is no element a a in A A which can be written as an average of a subset of A A not containing a a . We show that the largest non-averaging subset of \1, … , n\ { 1 , … , n } has size n1/4+o(1) 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 d -dimensional box for any fixed d 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.