Density theorems for anisotropic point configurations

TL;DR

This paper extends Euclidean density theorems to anisotropic point configurations, demonstrating that sets of positive density contain large anisotropically dilated configurations, with implications for harmonic analysis and geometric combinatorics.

Contribution

It introduces nonisotropic power-type density theorems for various point configurations, generalizing prior isotropic results and highlighting anisotropic multilinear singular integrals.

Findings

Proved density theorems for anisotropic dilations of simplices.

Extended results to anisotropic rectangular boxes and distance trees.

Linked combinatorial geometry with anisotropic multilinear operators.

Abstract

Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, here we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.