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

Durcik, Polona and Kovač, Vjekoslav (2021) Boxes, extended boxes and sets of positive upper density in the Euclidean space. Mathematical Proceedings of the Cambridge Philosophical Society, 171 (3). pp. 481-501. ISSN 0305-0041. doi:10.1017/s0305004120000316. https://resolver.caltech.edu/CaltechAUTHORS:20220104-413398800

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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: 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1017/s0305004120000316DOIArticle
https://arxiv.org/abs/1809.08692arXivDiscussion Paper
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).
Funders:
Funding AgencyGrant Number
Croatian Science FoundationUIP-2017-05-4129
Issue or Number:3
Classification Code:2020 Mathematics Subject Classification: Primary: 05D10, 28A75; Secondary: 11B30, 42B20
DOI:10.1017/s0305004120000316
Record Number:CaltechAUTHORS:20220104-413398800
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20220104-413398800
Official Citation:Durcik, P., & Kovač, V. (2021). Boxes, extended boxes and sets of positive upper density in the Euclidean space. Mathematical Proceedings of the Cambridge Philosophical Society, 171(3), 481-501. doi:10.1017/S0305004120000316
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:112662
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:04 Jan 2022 19:51
Last Modified:04 Jan 2022 19:51

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