Quantitative concatenation for polynomial box norms

TL;DR

This paper develops advanced techniques using PET and concatenation to control box norms in polynomial progressions, aiming to improve bounds on sets lacking such progressions in multidimensional settings.

Contribution

It introduces a method for establishing box-norm control with expected directions for polynomial progressions, advancing the understanding of their structure in higher dimensions.

Findings

Established box-norm control with expected directions for polynomial progressions.

Achieved polynomial losses in parameters, enabling explicit bounds.

Set the stage for bounding sets lacking polynomial progressions in multidimensional grids.

Abstract

Using PET and quantitative concatenation techniques, we establish box-norm control with the "expected" directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper arXiv:2407.08637, we complete this program for sets in $[N]^2$ lacking nondegenerate progressions of the form $(x, y), (x + P(z), y), (x, y + P(z))$, where $P \in \mathbb{Z}[z]$ is any fixed polynomial with an integer root of multiplicity $1$.