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arXiv · 2007.06005

On arithmetic progressions in non-periodic self-affine tilings

Yasushi Nagai·Shigeki Akiyama·Jeong-Yup Lee
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Abstract

We study the repetition of patches in self-affine tilings in 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/non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in R^d.

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Yasushi Nagai, Shigeki Akiyama, Jeong-Yup Lee. 2020-07-12. On arithmetic progressions in non-periodic self-affine tilings. https://doi.org/10.1017/etds.2021.59

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