Good bounds for sets lacking skew corners
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909168951099392 |
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| author | Milićević, Luka |
| author_facet | Milićević, Luka |
| contents | A skew corner is a triple of points in $\mathbb{Z} \times \mathbb{Z}$ of the form $(x,y), (x, y + a)$ and $(x + a, y')$. Pratt posed the following question: how large can a set $A \subseteq [n] \times [n]$ be, provided it contains no non-trivial skew corner (i.e. one for which $a\not=0$)? We prove that $|A| \leq \exp(- c\log^c n) n^2$, for an absolute constant $c > 0$, which, along with a construction of Beker, essentially resolves Pratt's question.
Our argument is represents a two-dimensional variant of the method of Kelley and Meka, which they used to prove Behrend-type bounds in Roth's theorem. A very similar result was obtained independently and simultaneously by Jaber, Lovett and Ostuni. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07180 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Good bounds for sets lacking skew corners Milićević, Luka Combinatorics Number Theory A skew corner is a triple of points in $\mathbb{Z} \times \mathbb{Z}$ of the form $(x,y), (x, y + a)$ and $(x + a, y')$. Pratt posed the following question: how large can a set $A \subseteq [n] \times [n]$ be, provided it contains no non-trivial skew corner (i.e. one for which $a\not=0$)? We prove that $|A| \leq \exp(- c\log^c n) n^2$, for an absolute constant $c > 0$, which, along with a construction of Beker, essentially resolves Pratt's question. Our argument is represents a two-dimensional variant of the method of Kelley and Meka, which they used to prove Behrend-type bounds in Roth's theorem. A very similar result was obtained independently and simultaneously by Jaber, Lovett and Ostuni. |
| title | Good bounds for sets lacking skew corners |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2404.07180 |