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Quantitative bounds in the nonlinear Roth theorem

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Quantitative bounds in the nonlinear Roth theorem. / Peluse, S.; Prendiville, S.
In: Inventiones Mathematicae, Vol. 238, No. 3, 31.12.2024, p. 865-903.

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

Harvard

Peluse, S & Prendiville, S 2024, 'Quantitative bounds in the nonlinear Roth theorem', Inventiones Mathematicae, vol. 238, no. 3, pp. 865-903. https://doi.org/10.1007/s00222-024-01293-x

APA

Peluse, S., & Prendiville, S. (2024). Quantitative bounds in the nonlinear Roth theorem. Inventiones Mathematicae, 238(3), 865-903. https://doi.org/10.1007/s00222-024-01293-x

Vancouver

Peluse S, Prendiville S. Quantitative bounds in the nonlinear Roth theorem. Inventiones Mathematicae. 2024 Dec 31;238(3):865-903. Epub 2024 Oct 7. doi: 10.1007/s00222-024-01293-x

Author

Peluse, S. ; Prendiville, S. / Quantitative bounds in the nonlinear Roth theorem. In: Inventiones Mathematicae. 2024 ; Vol. 238, No. 3. pp. 865-903.

Bibtex

@article{2f2a373f58d84b27a55edd3aba1bc5ee,
title = "Quantitative bounds in the nonlinear Roth theorem",
abstract = "We show that there exists c>0 such that any subset of {1,…,N} of density at least (loglogN) −c contains a nontrivial progression of the form x, x+y, x+y 2. This is the first quantitatively effective version of the Bergelson–Leibman polynomial Szemer{\'e}di theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations x, x+y, x+y 2 deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and Ziegler, with polynomial bounds.",
author = "S. Peluse and S. Prendiville",
year = "2024",
month = dec,
day = "31",
doi = "10.1007/s00222-024-01293-x",
language = "English",
volume = "238",
pages = "865--903",
journal = "Inventiones Mathematicae",
issn = "0020-9910",
publisher = "Springer New York",
number = "3",

}

RIS

TY - JOUR

T1 - Quantitative bounds in the nonlinear Roth theorem

AU - Peluse, S.

AU - Prendiville, S.

PY - 2024/12/31

Y1 - 2024/12/31

N2 - We show that there exists c>0 such that any subset of {1,…,N} of density at least (loglogN) −c contains a nontrivial progression of the form x, x+y, x+y 2. This is the first quantitatively effective version of the Bergelson–Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations x, x+y, x+y 2 deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and Ziegler, with polynomial bounds.

AB - We show that there exists c>0 such that any subset of {1,…,N} of density at least (loglogN) −c contains a nontrivial progression of the form x, x+y, x+y 2. This is the first quantitatively effective version of the Bergelson–Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations x, x+y, x+y 2 deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and Ziegler, with polynomial bounds.

U2 - 10.1007/s00222-024-01293-x

DO - 10.1007/s00222-024-01293-x

M3 - Journal article

VL - 238

SP - 865

EP - 903

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -