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ON HIGHER DIMENSIONAL ARITHMETIC PROGRESSIONS IN MEYER SETS

ON HIGHER DIMENSIONAL ARITHMETIC PROGRESSIONS IN MEYER SETS Abstract In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by . We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the Australian Mathematical Society Cambridge University Press

ON HIGHER DIMENSIONAL ARITHMETIC PROGRESSIONS IN MEYER SETS

25 pages

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References (35)

Publisher
Cambridge University Press
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
ISSN
1446-8107
eISSN
1446-7887
DOI
10.1017/S1446788721000215
Publisher site
See Article on Publisher Site

Abstract

Abstract In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by . We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.

Journal

Journal of the Australian Mathematical SocietyCambridge University Press

Published: Jun 1, 2023

Keywords: 11B25; 52C23; Meyer sets; model sets; cut-and-project schemes; arithmetic progressions; rank

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