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We generalize a result by Cook  et al . [ 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   $1&lt;p&lt;\infty$  ,   $p\neq 2$  , 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 ].

On Side Lengths of Corners in Positive Density Subsets of the Euclidean Space