On arithmetic progressions in non-periodic self-affine tilings
On arithmetic progressions in non-periodic self-affine tilings
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DOI：
10.1017/etds.2021.59
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发表时间：
2020-07
影响因子：
0.9
通讯作者：
Yasushi Nagai;S. Akiyama;Jeong-Yup Lee
中科院分区：
文献类型：
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作者：
Yasushi Nagai;S. Akiyama;Jeong-Yup Lee
摘要 我们研究 ${\mathbb {R}}^d$ 中自仿射平铺中补丁的重复。特别是，我们研究算术级数的存在和不存在。我们首先证明自仿射平铺展开图的算术条件意味着某些一维算术级数不存在。接下来，我们证明满秩无限算术级数、纯离散动力谱和极限周期性的存在对于某一类自仿射平铺来说都是等价的。最后，我们给出了 ${\mathbb {R}}^d$ 中自相似平铺中是否存在满秩无限算术级数的完整图像。
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 ${\mathbb {R}}^d$ .