Bounds in a popular multidimensional nonlinear Roth theorem
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914866450661376 |
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| author | Peluse, Sarah Prendiville, Sean Shao, Xuancheng |
| author_facet | Peluse, Sarah Prendiville, Sean Shao, Xuancheng |
| contents | A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective ``popular'' version of this result, showing that every dense set has some non-zero $d$ such that the number of configurations with difference parameter $d$ is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower-type bounds encountered in the popular linear Roth theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_08338 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds in a popular multidimensional nonlinear Roth theorem Peluse, Sarah Prendiville, Sean Shao, Xuancheng Number Theory Combinatorics A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective ``popular'' version of this result, showing that every dense set has some non-zero $d$ such that the number of configurations with difference parameter $d$ is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower-type bounds encountered in the popular linear Roth theorem. |
| title | Bounds in a popular multidimensional nonlinear Roth theorem |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2407.08338 |