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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: 2n 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.


This article was originally published in Mathematical Proceedings of the Cambridge Philosophical Society, volume 171, issue 3, in 2021.

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This work is licensed under a Creative Commons Attribution 4.0 License.



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