Strong Bounds for 3-Progressions
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914994018320384 |
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| author | Kelley, Zander Meka, Raghu |
| author_facet | Kelley, Zander Meka, Raghu |
| contents | We show that for some constant $β> 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^β)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic progressions were known to exist only for sets of size at least $N/(\log N)^{1 + c}$ for a constant $c > 0$.
Our approach is first to develop new analytic techniques for addressing some related questions in the finite-field setting and then to apply some analogous variants of these same techniques, suitably adapted for the more complicated setting of integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05537 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strong Bounds for 3-Progressions Kelley, Zander Meka, Raghu Number Theory Combinatorics We show that for some constant $β> 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^β)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic progressions were known to exist only for sets of size at least $N/(\log N)^{1 + c}$ for a constant $c > 0$. Our approach is first to develop new analytic techniques for addressing some related questions in the finite-field setting and then to apply some analogous variants of these same techniques, suitably adapted for the more complicated setting of integers. |
| title | Strong Bounds for 3-Progressions |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2302.05537 |