Sharp Bound for the Erdős–Straus Non-averaging Set Problem
Creators
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1.
California Institute of Technology
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2.
Massachusetts Institute of Technology
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.
Copyright and License
© The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funding
Pham’s research is supported by a Clay Research Fellowship. Zakharov’s research was supported by the Jane Street Graduate Fellowship.
Open Access funding provided by the MIT Libraries.
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Additional details
Related works
- Describes
- Journal Article: https://rdcu.be/eUhfo (ReadCube)
- Is new version of
- Discussion Paper: arXiv:2410.14624 (arXiv)
Funding
- Clay Mathematics Institute
- Massachusetts Institute of Technology
Dates
- Submitted
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2024-11-25
- Accepted
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2025-11-17
- Available
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2025-12-03Published