2 research outputs found
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 to corners in subsets of
. More precisely, if , ,
and is large enough, we show that an arbitrary measurable set
of positive upper Banach density
contains corners , , such that the -norm of
the side 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].Comment: 17 pages; v2: several computations expanded, references added and
update