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

Creators: Durcik, Polona; Kovač, Vjekoslav

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: 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.

Additional Information

© Cambridge Philosophical Society 2021. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Received 11 March 2019; revised 27 August 2020. First published online 14 January 2021. The authors thank Christoph Thiele for inspiring discussions aided by the bilateral DAAD-MZO grant Multilinear singular integrals and applications. The authors are also grateful to the anonymous referee for useful suggestions. V. K. was supported in part by the Croatian Science Foundation under the project UIP-2017-05-4129 (MUNHANAP).

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