Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916534459301888 |
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| author | Bhangale, Amey Khot, Subhash Minzer, Dor |
| author_facet | Bhangale, Amey Khot, Subhash Minzer, Dor |
| contents | For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq \mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such a set is at most $\frac{C}{(\log\log\log n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/\log^{*} n)$, which follows from the density Hales-Jewett theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06600 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$ Bhangale, Amey Khot, Subhash Minzer, Dor Combinatorics Discrete Mathematics For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq \mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such a set is at most $\frac{C}{(\log\log\log n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/\log^{*} n)$, which follows from the density Hales-Jewett theorem. |
| title | Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$ |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2308.06600 |