- https://arxiv.org/abs/1903.02592
Submitted manuscript

Research output: Contribution to journal › Journal article

Published

**Quantitative bounds in the non-linear Roth theorem.** / Peluse, Sarah; Prendiville, Sean.

Research output: Contribution to journal › Journal article

@article{2fdcf65d3f8748718f72dd86321fd2b0,

title = "Quantitative bounds in the non-linear Roth theorem",

abstract = "We show that any subset of $[N]$ of density at least $(\log\log{N})^{-2^{-157}}$ 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\'edi theorem for a progression involving polynomials of differing degrees. In the course of the proof, we also develop a quantitative version of a special case of a concatenation theorem of Tao and Ziegler, with polynomial bounds.",

author = "Sarah Peluse and Sean Prendiville",

year = "2019",

month = mar

day = "6",

language = "English",

journal = "arXiv",

}

TY - JOUR

T1 - Quantitative bounds in the non-linear Roth theorem

AU - Peluse, Sarah

AU - Prendiville, Sean

PY - 2019/3/6

Y1 - 2019/3/6

N2 - We show that any subset of $[N]$ of density at least $(\log\log{N})^{-2^{-157}}$ 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\'edi theorem for a progression involving polynomials of differing degrees. In the course of the proof, we also develop a quantitative version of a special case of a concatenation theorem of Tao and Ziegler, with polynomial bounds.

AB - We show that any subset of $[N]$ of density at least $(\log\log{N})^{-2^{-157}}$ 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\'edi theorem for a progression involving polynomials of differing degrees. In the course of the proof, we also develop a quantitative version of a special case of a concatenation theorem of Tao and Ziegler, with polynomial bounds.

M3 - Journal article

JO - arXiv

JF - arXiv

ER -