Constructions of normal numbers with infinitely many digits
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909116891398144 |
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| author | Boonstra, Aafko Kalle, Charlene |
| author_facet | Boonstra, Aafko Kalle, Charlene |
| contents | Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the countably infinite alphabet $\mathbb N$ in which each possible block of digits $α_1, \ldots, α_k \in \mathbb N$, $k \in \mathbb N$, occurs with frequency $\prod_{d=1}^k L_{α_d}$. In other words, we construct $L$-normal sequences. These sequences can then be projected to normal numbers in various affine number systems, such as real numbers $x \in [0,1]$ that are normal in GLS number systems that correspond to the sequence $L$ or higher dimensional variants. In particular, this construction provides a family of numbers that have a normal Lüroth expansion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructions of normal numbers with infinitely many digits Boonstra, Aafko Kalle, Charlene Number Theory 11K16, 11A63, 05C05 Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the countably infinite alphabet $\mathbb N$ in which each possible block of digits $α_1, \ldots, α_k \in \mathbb N$, $k \in \mathbb N$, occurs with frequency $\prod_{d=1}^k L_{α_d}$. In other words, we construct $L$-normal sequences. These sequences can then be projected to normal numbers in various affine number systems, such as real numbers $x \in [0,1]$ that are normal in GLS number systems that correspond to the sequence $L$ or higher dimensional variants. In particular, this construction provides a family of numbers that have a normal Lüroth expansion. |
| title | Constructions of normal numbers with infinitely many digits |
| topic | Number Theory 11K16, 11A63, 05C05 |
| url | https://arxiv.org/abs/2402.14500 |