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Quantitative bounds in the non-linear Roth theorem

Research output: Contribution to journalJournal article

Published
<mark>Journal publication date</mark>6/03/2019
<mark>Journal</mark>arXiv
Publication statusPublished
Original languageEnglish

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.