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arXiv · 1809.08692

Boxes, extended boxes, and sets of positive upper density in the Euclidean space

Polona Durcik·Vjekoslav Kovač
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Abstract

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed $n$-dimensional rectangular box, the same vertices extended with $n$ points completing three-term arithmetic progressions, and the same vertices extended with $n$ points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

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Polona Durcik, Vjekoslav Kovač. 2020-08-27. Boxes, extended boxes, and sets of positive upper density in the Euclidean space. https://doi.org/10.1017/s0305004120000316

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