A Polylogarithmic Bound in the Nonlinear Roth Theorem

Abstract

We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^2$ have at most polylogarithmic density.

Bounds for sets with no polynomial progressions

“…In the general situation covered by Theorem 1.1, these averages of Gowers box norms can become arbitrarily complex, necessitating a new and more general approach. The concatenation theory developed in this article is significantly stronger than that in [15], and the bulk of the new ideas in this article go into proving these concatenation results. We must also be more careful during the PET induction step than in previous works in order to produce an average of Gowers box norms of the particular form that our concatenation result can be applied to.…”

“…The proof of Theorem 1.1 involves adapting the central idea of [14] to the integer setting. Such an adaptation was first done by Prendiville and the author [15] for the special case of the progression , + , + 2 , showing that , 2 ( ) ≪ /(log log ) for some absolute constant > 0. It turns out that the assumption that 1 , .…”

“…We now briefly discuss the proof of Theorem 1.1 in comparison to the arguments in [14] and [15]. The proof of Theorem 1.1 proceeds via a density increment argument where, as in [15], it is shown that any subset of [ ] with no nontrivial polynomial progressions has increased density on a long arithmetic progression with very small common difference.…”

“…We now briefly discuss the proof of Theorem 1.1 in comparison to the arguments in [14] and [15]. The proof of Theorem 1.1 proceeds via a density increment argument where, as in [15], it is shown that any subset of [ ] with no nontrivial polynomial progressions has increased density on a long arithmetic progression with very small common difference. This is done by following the strategy for proving quantitative bounds in the polynomial Szemerédi theorem originating in [14], which is to first show that the count of polynomial progressions in a set is controlled by some Gowers -norm and then to show that, in certain situations, one can combine this -control with understanding of shorter progressions to deduce −1 -control.…”

“…In this article, we prove a new quantitative concatenation result, which we use to control (with polynomial bounds) the averages of Gowers box norms just mentioned by anorm for some depending only on the degrees of the polynomials involved. In [15], this was done for the special case of the average of Gowers box norms controlling the progression , + , + 2 , which is the simplest case requiring a nontrivial concatenation argument. In the general situation covered by Theorem 1.1, these averages of Gowers box norms can become arbitrarily complex, necessitating a new and more general approach.…”