On side lengths of corners in positive density subsets of the Euclidean space

TL;DR

This paper extends results on arithmetic progressions to corners in high-dimensional Euclidean spaces, showing that positive density sets contain corners with side lengths covering all large values, using advanced harmonic analysis techniques.

Contribution

It introduces a higher-dimensional multilinear estimate to prove the existence of corners with large side lengths in positive density subsets of Euclidean space.

Findings

Positive density sets contain corners with arbitrarily large side lengths.

The proof involves a novel higher-dimensional multilinear harmonic analysis estimate.

Extension of prior results on arithmetic progressions to geometric configurations.

Abstract

We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of $\mathbb{R}^d\times\mathbb{R}^d$. More precisely, if $1<p<\infty$, $p\neq 2$, and $d$ is large enough, we show that an arbitrary measurable set $A\subseteq\mathbb{R}^d\times\mathbb{R}^d$ of positive upper Banach density contains corners $(x,y)$, $(x+s,y)$, $(x,y+s)$ such that the $\ell^p$-norm of the side $s$ attains all sufficiently large real values. Even though we closely follow the basic steps from [3], the proof diverges at the part relying on harmonic analysis. We need to apply a higher-dimensional variant of a multilinear estimate from [5], which we establish using the techniques from [5] and [6].