Quantitative bounds in a popular polynomial Szemer\'{e}di theorem

TL;DR

This paper establishes polylogarithmic bounds for polynomial Szemerédi theorems with polynomials of distinct degrees and zero constant terms, demonstrating that dense sets contain many polynomial progressions.

Contribution

It provides the first polylogarithmic bounds for polynomial Szemerédi theorems under specified conditions and introduces an effective 'popular' version for dense subsets.

Findings

Dense sets with logarithmic density contain polynomial progressions.

Polylogarithmic bounds depend on polynomial degrees and zero constant terms.

A 'popular' version shows many progressions for some fixed difference.

Abstract

We obtain polylogarithmic bounds in the polynomial Szemer\'{e}di theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each having zero constant term. Then there exists a constant $c = c(P_1,\dots,P_m) > 0$ such that any subset $A \subset \{1,2,\dots,N\}$ of density at least $(\log N)^{-c}$ contains a nontrivial polynomial progression of the form $x, x+P_1(y), \dots, x+P_m(y)$. In addition, we prove an effective ``popular'' version, showing that every dense subset $A$ has some non-zero $y$ such that the number of polynomial progressions in $A$ with this difference $y$ is asymptotically at least as large as in a random set of the same density as $A$.