Polynomial progressions in topological fields

TL;DR

This paper establishes a quantitative count of polynomial progressions within sets of positive density in topological fields, utilizing a novel inverse theorem and Sobolev inequalities applicable across various analysis contexts.

Contribution

It introduces a general $L^{inity}$ inverse theorem and Sobolev improving estimates for multilinear polynomial averages in topological fields, advancing the understanding of polynomial progressions.

Findings

Quantitative count of polynomial progressions in topological fields

Development of a new inverse theorem of independent interest

Sobolev inequalities applicable in real, complex, and p-adic analysis

Abstract

Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y), \ldots, x + P_m(y)$ lying in a set $S\subseteq K$ of positive density. The proof relies on a general $L^{\infty}$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and $p$-adic analysis.