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On arithmetic progressions in non-periodic self-affine tilings

Part of: Discrete geometry

Published online by Cambridge University Press:  15 June 2021

YASUSHI NAGAI ,
SHIGEKI AKIYAMA  and
JEONG-YUP LEE
YASUSHI NAGAI*
Affiliation:
School of General Education, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano390-8621, Japan
SHIGEKI AKIYAMA
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki305-8571, Japan (e-mail: akiyama@math.tsukuba.ac.jp)
JEONG-YUP LEE
Affiliation:
Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701, Korea, or KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul02455, Korea (e-mail: jylee@cku.ac.kr)
*
e-mail: ynagai@shinshu-u.ac.jp
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Abstract

We study the repetition of patches in self-affine tilings in ${\mathbb {R}}^d$ . In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit-periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in ${\mathbb {R}}^d$ .

Keywords

tilingtiling dynamical systemdynamical spectrumarithmetic progression

MSC classification

Secondary: 52C23: Quasicrystals, aperiodic tilings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2957 - 2989
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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