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Postfach 100131, 33501 Bielefeld Germany e-mail: anna.klick@uni-bielefeld.de NICOLAE STRUNGARU
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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 of the Australian Mathematical Society – Cambridge University Press
Published: Jun 1, 2023
Keywords: 11B25; 52C23; Meyer sets; model sets; cut-and-project schemes; arithmetic progressions; rank
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