---
abstract: 'We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^2$ have at most polylogarithmic density.'
address:
- |
    Mathematical Institute\
    University of Oxford\
    UK
- |
    Department of Mathematics and Statistics\
    Lancaster University\
    UK
author:
- Sarah Peluse
- Sean Prendiville
bibliography:
- '/Users/sean/Dropbox/SeanBib.bib'
title: A polylogarithmic bound in the nonlinear Roth theorem
---

Introduction
============

Statement of results
--------------------

In [@PelusePrendivilleQuantitative] the authors obtained, for the first time, an effective bound for subsets of ${\left\{1, \dots, N\right\}}$ lacking the nonlinear Roth configuration $x$, $x+y$, $x+y^2$. There it was established that such sets have cardinality at most $O(N/(\log\log N)^c)$, where $c> 0$ is an absolute constant. The key breakthrough of [@PelusePrendivilleQuantitative] was a “local $U^1$-control” result, from which a bound for sets lacking the nonlinear Roth configuration follows via standard methods. Here, we combine this local $U^1$-control result with a more sophisticated argument to remove a logarithm from the bound of [@PelusePrendivilleQuantitative].

\[main\] There exists an absolute constant $c > 0$ such that the following holds. Suppose that $A \subset {\left\{1,\dots, N\right\}}$ lacks configurations of the form $$\label{main config}
x,\ x+y, \ x+y^2 \qquad (y \neq 0).$$ Then$$|A| = O\left( N/(\log N)^{c}\right).$$

A careful analysis shows 