On polynomial progressions via transference

TL;DR

This paper establishes new bounds for polynomial progressions in Szemerédi's theorem, demonstrating that sets avoiding such patterns are very small, with bounds involving iterated logarithms, over both finite fields and integers.

Contribution

It provides the first reasonable bounds for polynomial Szemerédi theorem in both finite cyclic groups and integers, especially for fixed polynomial differences.

Findings

Sets avoiding polynomial progressions are extremely small, bounded by a poly-logarithmic factor.

The bounds are valid over both finite fields and the integers.

The results extend the understanding of polynomial configurations in additive combinatorics.

Abstract

We prove new cases of reasonable bounds for the polynomial Szemer\'{e}di theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemer\'edi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\in \mathbb{Z}[y]$ with $P(0) = 0$, that the largest subset $A\subseteq [N]$ avoiding the pattern \[x, x+P(y),\ldots, x+ kP(y)\] has size bounded by $\ll_{P,k}N(\log\log\log N)^{-\Omega_{P,k}(1)}.$