An inverse theorem on sets with rich additive structure modulo primes
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911301121343488 |
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| author | Croot, Ernie Mao, Junzhe Yip, Chi Hoi |
| author_facet | Croot, Ernie Mao, Junzhe Yip, Chi Hoi |
| contents | In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in an interval $[y,2y]$ with $(\log N)^{O(1)} < y \leq N$. We also provide several constructions demonstrating the sharpness of our results. Furthermore, as an application, we provide several improvements on the larger sieve bound for $|S|$ when $S$ mod $p$ has strong additive structure, parallel to the work of Green--Harper and Shao for improvements on the large sieve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An inverse theorem on sets with rich additive structure modulo primes Croot, Ernie Mao, Junzhe Yip, Chi Hoi Number Theory Combinatorics Primary 11N35, 11N69, Secondary 11B30, 11P70 In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in an interval $[y,2y]$ with $(\log N)^{O(1)} < y \leq N$. We also provide several constructions demonstrating the sharpness of our results. Furthermore, as an application, we provide several improvements on the larger sieve bound for $|S|$ when $S$ mod $p$ has strong additive structure, parallel to the work of Green--Harper and Shao for improvements on the large sieve. |
| title | An inverse theorem on sets with rich additive structure modulo primes |
| topic | Number Theory Combinatorics Primary 11N35, 11N69, Secondary 11B30, 11P70 |
| url | https://arxiv.org/abs/2510.08862 |