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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2401.10574 |
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| _version_ | 1866913200534978560 |
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| author | Li, Qian Rao, Hui |
| author_facet | Li, Qian Rao, Hui |
| contents | Let $R$ be an $n\times n$ expanding matrix with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set $\mathcal{D}\subset\mathbb{Z}^n$ so that the integral self-affine set $T(R,\mathcal{D})$ is a translational tile on $\mathbb{R}^n$ In this paper, we introduce a notion of skew-product-form digit set which is a very general class of tile digit sets. Especially, we show that in the one-dimensional case, if $T(b,\mathcal{D})$ is a self-similar tile, then there exists $m\geq 1$ such that $$\mathcal{D}_m=\mathcal{D}+b\mathcal{D}+\cdots+b^{m-1}\mathcal{D}$$ is a skew-product-form digit set. Notice that $T(b,\mathcal{D})=T(b^m,\mathcal{D}_m)$, in some sense, we completely characterize the self-similar tiles in $\mathbb{R}^1$ As an application, we establish that all self-similar tiles $T(b,\mathcal{D})$ where $b=p^αq^β$ contains at most two prime factors are spectral sets in $\mathbb{R}^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of self-affine tile digit sets on $\mathbb{R}^n$ Li, Qian Rao, Hui Number Theory Let $R$ be an $n\times n$ expanding matrix with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set $\mathcal{D}\subset\mathbb{Z}^n$ so that the integral self-affine set $T(R,\mathcal{D})$ is a translational tile on $\mathbb{R}^n$ In this paper, we introduce a notion of skew-product-form digit set which is a very general class of tile digit sets. Especially, we show that in the one-dimensional case, if $T(b,\mathcal{D})$ is a self-similar tile, then there exists $m\geq 1$ such that $$\mathcal{D}_m=\mathcal{D}+b\mathcal{D}+\cdots+b^{m-1}\mathcal{D}$$ is a skew-product-form digit set. Notice that $T(b,\mathcal{D})=T(b^m,\mathcal{D}_m)$, in some sense, we completely characterize the self-similar tiles in $\mathbb{R}^1$ As an application, we establish that all self-similar tiles $T(b,\mathcal{D})$ where $b=p^αq^β$ contains at most two prime factors are spectral sets in $\mathbb{R}^1$. |
| title | Characterization of self-affine tile digit sets on $\mathbb{R}^n$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.10574 |