Sharp bound for the Erdős-Straus non-averaging set problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915487823167488 |
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| author | Pham, Huy Tuan Zakharov, Dmitrii |
| author_facet | Pham, Huy Tuan Zakharov, Dmitrii |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp bound for the Erdős-Straus non-averaging set problem Pham, Huy Tuan Zakharov, Dmitrii Combinatorics Number Theory 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 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. |
| title | Sharp bound for the Erdős-Straus non-averaging set problem |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2410.14624 |