Integer tile and Spectrality of Cantor-Moran measures with equidifferent digit sets
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914994730303488 |
|---|---|
| author | Wu, Sha Xiao, Yingqing |
| author_facet | Wu, Sha Xiao, Yingqing |
| contents | Let $\left\{b_{k}\right\}_{k=1}^{\infty}$ be a sequence of integers with $|b_{k}|\geq2$ and $\left\{D_{k}\right\}_{k=1}^{\infty} $ be a sequence of equidifferent digit sets with $D_{k}=\left\{0,1, \cdots, N-1\right\}t_{k},$ where $N\geq2$ is a prime number and $\{t_{k}\}_{k=1}^{\infty}$ is bounded. In this paper, we study the existence of the Cantor-Moran measure $μ_{\{b_k\},\{D_k\}}$ and show that $$\mathbf{D}_k:=D_k\oplus b_{k} D_{k-1}\oplus b_{k}b_{k-1} D_{k-2}\oplus\cdots\oplus b_{k}b_{k-1}\cdots b_2D_{1}$$ is an integer tile for all $k\in\mathbb{N}^+$ if and only if $\mathbf{s}_i\neq\mathbf{s}_j$ for all $i\neq j\in\mathbb{N}^{+}$, where $\mathbf{s}_i$ is defined as the numbers of factor $N$ in $\frac{b_1b_2\cdots b_i}{Nt_i}$. Moreover, we prove that $\mathbf{D}_k$ being an integer tile for all $k\in\mathbb{N}^+$ is a necessary condition for the Cantor-Moran measure to be a spectral measure, and we provide an example to demonstrate that it cannot become a sufficient condition. Furthermore, under some additional assumptions, we establish that the Cantor-Moran measure to be a spectral measure is equivalent to $\mathbf{D}_k$ being an integer tile for all $k\in\mathbb{N}^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21626 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integer tile and Spectrality of Cantor-Moran measures with equidifferent digit sets Wu, Sha Xiao, Yingqing Number Theory Classical Analysis and ODEs Primary 28A25, 28A80, Secondary 42C05, 46C05 Let $\left\{b_{k}\right\}_{k=1}^{\infty}$ be a sequence of integers with $|b_{k}|\geq2$ and $\left\{D_{k}\right\}_{k=1}^{\infty} $ be a sequence of equidifferent digit sets with $D_{k}=\left\{0,1, \cdots, N-1\right\}t_{k},$ where $N\geq2$ is a prime number and $\{t_{k}\}_{k=1}^{\infty}$ is bounded. In this paper, we study the existence of the Cantor-Moran measure $μ_{\{b_k\},\{D_k\}}$ and show that $$\mathbf{D}_k:=D_k\oplus b_{k} D_{k-1}\oplus b_{k}b_{k-1} D_{k-2}\oplus\cdots\oplus b_{k}b_{k-1}\cdots b_2D_{1}$$ is an integer tile for all $k\in\mathbb{N}^+$ if and only if $\mathbf{s}_i\neq\mathbf{s}_j$ for all $i\neq j\in\mathbb{N}^{+}$, where $\mathbf{s}_i$ is defined as the numbers of factor $N$ in $\frac{b_1b_2\cdots b_i}{Nt_i}$. Moreover, we prove that $\mathbf{D}_k$ being an integer tile for all $k\in\mathbb{N}^+$ is a necessary condition for the Cantor-Moran measure to be a spectral measure, and we provide an example to demonstrate that it cannot become a sufficient condition. Furthermore, under some additional assumptions, we establish that the Cantor-Moran measure to be a spectral measure is equivalent to $\mathbf{D}_k$ being an integer tile for all $k\in\mathbb{N}^+$. |
| title | Integer tile and Spectrality of Cantor-Moran measures with equidifferent digit sets |
| topic | Number Theory Classical Analysis and ODEs Primary 28A25, 28A80, Secondary 42C05, 46C05 |
| url | https://arxiv.org/abs/2410.21626 |