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

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 $1<p<\mathrm{\infty }$, $p\mathrm{\ne }2$, 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 ${\mathrm{\ell }}^p$-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].