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arXiv · 2407.08338

Bounds in a popular multidimensional nonlinear Roth theorem

Sarah Peluse·Sean Prendiville·Xuancheng Shao
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Abstract

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 Szemer\'edi 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.

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Sarah Peluse, Sean Prendiville, Xuancheng Shao. 2024-07-11. Bounds in a popular multidimensional nonlinear Roth theorem. https://arxiv.org/abs/2407.08338

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