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arXiv · 2410.14624

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

Huy Tuan Pham·Dmitrii Zakharov
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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.

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Huy Tuan Pham, Dmitrii Zakharov. 2024-10-18. Sharp bound for the Erd\H{o}s-Straus non-averaging set problem. https://arxiv.org/abs/2410.14624

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