Bounds for sets with no polynomial progressions

TL;DR

This paper establishes upper bounds on the size of subsets of integers avoiding polynomial progressions, demonstrating they are significantly smaller than the entire set, and introduces a general method linking polynomial progression counts to Gowers norms.

Contribution

It provides new bounds for sets lacking polynomial progressions and develops a general framework connecting progression counts with Gowers norms.

Findings

Sets with no polynomial progressions are very sparse, bounded by N divided by a polylogarithmic factor.

A general result controlling polynomial progression counts via Gowers norms is proved.

The bounds depend on the degrees of the polynomials involved.

Abstract

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset $A$ of $\{1,\dots,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots,x+P_m(y)$ has size $|A|\ll N/(\log\log{N})^{c_{P_1,\dots,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.