Quantitative bounds in a popular polynomial Szemerédi theorem
Xuancheng Shao, Mengdi Wang
TL;DR
The paper advances quantitative bounds for the polynomial Szemerédi theorem in the finite setting, focusing on distinct-degree polynomials with zero constant terms. It develops a multi-faceted inverse-theorem framework and a local-factor regularity scheme to translate large configuration counts into correlations with structured Lipschitz phases, enabling polylogarithmic density bounds and an effective popular-difference result. The key innovations include an inductive argument that propagates correlation across all participating polynomials, a decomposition-based transition from single-function to multi-function correlations, and a local-factor machinery compatible with a Heath–Brown–Szemerédi style density increment. Together, these results deepen the understanding of polynomial configurations in dense sets and provide effective, explicit bounds with potential implications for further quantitative progress in polynomial Szemerédi-type problems.
Abstract
We obtain polylogarithmic bounds in the polynomial Szemerédi 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 $A$.