On long arithmetic progressions in binary Morse-like words

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.

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“…Moreover, we showed that van der Waerden-type theorems hold in Meyer sets. More recently, related results have been investigated in [1,16].…”

Section: Introductionmentioning

confidence: 99%

On Higher Dimensional Arithmetic Progressions in Meyer Sets

Klick¹,

Strungaru

2021

J. Aust. Math. Soc.

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show abstract

“…Moreover, we showed that van der Waerden type theorems hold in Meyer sets. More recently, related results have been investigated in [1,21].…”

Section: Introductionmentioning

confidence: 99%

On higher dimensional arithmetic progressions in Meyer sets

Klick¹,

Strungaru²

2021

Preprint

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

show abstract

On long arithmetic progressions in binary Morse-like words

Cited by 2 publications

References 24 publications

On Higher Dimensional Arithmetic Progressions in Meyer Sets

On Higher Dimensional Arithmetic Progressions in Meyer Sets

On higher dimensional arithmetic progressions in Meyer sets

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