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

Abstract A set of integers $A$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is non-averaging if there is no element $a$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> </mml:math> in $A$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> which can be written as an average of a subset of $A$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> not containing $a$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> </mml:math> . We show that the largest non-averaging subset of $\{1, \ldots , n\}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:math> has size $n^{1/4+o(1)}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> , thus solving the Erdős–Straus problem. We also determine the largest size of a non-averaging set in a $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> -dimensional box for any fixed $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> . 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.