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

IF 2.2 1区数学 Q1 MATHEMATICS

Geometric and Functional Analysis Pub Date : 2025-12-03 DOI:10.1007/s00039-025-00728-8

Huy Tuan Pham, Dmitrii Zakharov

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引用次数: 0

Abstract

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

$\{1, \ldots , n\}$

{ 1 , … , n } has size

$n^{1/4+o(1)}$

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.

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Erdős-Straus非平均集问题的夏普界

一组整数$A$ A是非平均的，如果$A$ A$ A中没有元素$A$ A，它可以写成$A$ A的一个子集$A$ A不包含$A$ A的平均值。我们证明了$\{1,\ldots, n\}${1，…，n}的最大非平均子集的大小为$n^{1/4+o(1)}$ n 1/4+o(1)，从而解决了Erdős-Straus问题。对于任意固定的d$ d，我们还确定了d$ d维盒子中非平均集的最大大小。我们的主要工具包括Conlon， Fox和第一作者的子集和集的结构定理，以及关于近凸位置点集结构的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Geometric and Functional Analysis 数学-数学

CiteScore

3.70

自引率

4.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.