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Quantitative bounds in the non-linear Roth theorem

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

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Quantitative bounds in the non-linear Roth theorem. / Prendiville, Sean; Peluse, Sarah.
In: Inventiones Mathematicae, 20.09.2024.

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

Harvard

Prendiville, S & Peluse, S 2024, 'Quantitative bounds in the non-linear Roth theorem', Inventiones Mathematicae. <https://arxiv.org/abs/1903.02592>

APA

Prendiville, S., & Peluse, S. (in press). Quantitative bounds in the non-linear Roth theorem. Inventiones Mathematicae. https://arxiv.org/abs/1903.02592

Vancouver

Prendiville S, Peluse S. Quantitative bounds in the non-linear Roth theorem. Inventiones Mathematicae. 2024 Sept 20.

Author

Prendiville, Sean ; Peluse, Sarah. / Quantitative bounds in the non-linear Roth theorem. In: Inventiones Mathematicae. 2024.

Bibtex

@article{9fca17e44dd249e8bca5c35213b46009,
title = "Quantitative bounds in the non-linear 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 = "Sean Prendiville and Sarah Peluse",
year = "2024",
month = sep,
day = "20",
language = "English",
journal = "Inventiones Mathematicae",
issn = "0020-9910",
publisher = "Springer New York",

}

RIS

TY - JOUR

T1 - Quantitative bounds in the non-linear Roth theorem

AU - Prendiville, Sean

AU - Peluse, Sarah

PY - 2024/9/20

Y1 - 2024/9/20

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.

M3 - Journal article

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

ER -