A solution to the Erdős-Sárközy-Sós problem on asymptotic Sidon bases of order 3
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914791177584640 |
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| author | Pilatte, Cédric |
| author_facet | Pilatte, Cédric |
| contents | A set $S\subset \mathbb{N}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leq s_2$) are distinct. A set $S\subset \mathbb{N}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb{F}_q[t]$-analogue of Montgomery's conjecture for convolutions of the von Mangoldt function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09659 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A solution to the Erdős-Sárközy-Sós problem on asymptotic Sidon bases of order 3 Pilatte, Cédric Number Theory Combinatorics Primary 11B13, Secondary 11R58, 11N13, 05D40 A set $S\subset \mathbb{N}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leq s_2$) are distinct. A set $S\subset \mathbb{N}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb{F}_q[t]$-analogue of Montgomery's conjecture for convolutions of the von Mangoldt function. |
| title | A solution to the Erdős-Sárközy-Sós problem on asymptotic Sidon bases of order 3 |
| topic | Number Theory Combinatorics Primary 11B13, Secondary 11R58, 11N13, 05D40 |
| url | https://arxiv.org/abs/2303.09659 |