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

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 $1<p<\infty$, $p\ne 2$, 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].

17 pages; v2: several computations expanded, references added and updated

References in corpus (4)

• Norm-variation of ergodic averages with respect to two commuting transformations
• A Roth type theorem for dense subsets of ${\mathbb{R}}^d$
• An $L^4$ estimate for a singular entangled quadrilinear form
• Product of simplices and sets of positive upper density in ${\mathbb{R}}^d$

Cited by in corpus (10)

• A polynomial Roth theorem on the real line
• A Szemerédi-type theorem for subsets of the unit cube
• Boxes, extended boxes, and sets of positive upper density in the Euclidean space
• Density theorems for anisotropic point configurations
• A Non-Linear Roth Theorem for Sets of Positive Density
• The density of sets containing large similar copies of finite sets
• Gowers norms control diophantine inequalities
• T(1) theorem for dyadic singular integral forms associated with hypergraphs
• A Polynomial Roth Theorem for Corners in Finite Fields
• Large dilates of hypercube graphs in the plane