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

We generalize a result by Cook et al. [3] on three-term arithmetic progressions in subsets of Rd to corners in subsets of Rd×Rd⁠. More precisely, if 1<p<∞⁠, p≠2⁠, and d is large enough, we show that an arbitrary measurable set A⊆Rd×Rd of positive upper Banach density contains corners (x, y)⁠, (x+s, y)⁠, (x, y+s) such that the ℓp-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].

arithmetic progression, corner, multilinear estimate, Gowers norm

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