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Bounds in a popular multidimensional nonlinear Roth theorem

Research output: Contribution to Journal/Magazine › Journal article › peer-review

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Bounds in a popular multidimensional nonlinear Roth theorem. / Prendiville, Sean; Peluse, Sarah; Shao, Xuancheng.
In: Journal of the London Mathematical Society, 20.09.2024.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Prendiville, S, Peluse, S & Shao, X 2024, 'Bounds in a popular multidimensional nonlinear Roth theorem', Journal of the London Mathematical Society. <https://arxiv.org/abs/2407.08338>

APA

Prendiville, S., Peluse, S., & Shao, X. (in press). Bounds in a popular multidimensional nonlinear Roth theorem. Journal of the London Mathematical Society. https://arxiv.org/abs/2407.08338

Vancouver

Prendiville S, Peluse S, Shao X. Bounds in a popular multidimensional nonlinear Roth theorem. Journal of the London Mathematical Society. 2024 Sept 20.

Author

Prendiville, Sean ; Peluse, Sarah ; Shao, Xuancheng. / Bounds in a popular multidimensional nonlinear Roth theorem. In: Journal of the London Mathematical Society. 2024.

Bibtex

@article{f210e5711aae45279f80322231f1e447,
title = "Bounds in a popular multidimensional nonlinear Roth theorem",
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{\'e}di 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.",
author = "Sean Prendiville and Sarah Peluse and Xuancheng Shao",
year = "2024",
month = sep,
day = "20",
language = "English",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "Oxford University Press",

}

RIS

TY - JOUR

T1 - Bounds in a popular multidimensional nonlinear Roth theorem

AU - Prendiville, Sean

AU - Peluse, Sarah

AU - Shao, Xuancheng

PY - 2024/9/20

Y1 - 2024/9/20

N2 - 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.

AB - 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.

M3 - Journal article

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

ER -