An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912362041180160 |
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| author | Caalim, Jonathan Nollen, Nathaniel |
| author_facet | Caalim, Jonathan Nollen, Nathaniel |
| contents | In this paper, we consider the positional numeration system, called the Cantor real expansion, on the unit interval $[γ, γ+1]$, where $γ\in \mathbb{R}$, with respect to an alternate base (i.e., a base which is a purely periodic sequence of real numbers). In particular, we study the case where the expansion of $γ+1$ is periodic. Under certain assumptions, the base satisfies algebraic properties. We compute the bounds for the norms of the nontrivial Galois conjugates associated with the base; thereby, extending the results of Solomyak on the classical beta expansions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases Caalim, Jonathan Nollen, Nathaniel Number Theory 11A63, 11R09 In this paper, we consider the positional numeration system, called the Cantor real expansion, on the unit interval $[γ, γ+1]$, where $γ\in \mathbb{R}$, with respect to an alternate base (i.e., a base which is a purely periodic sequence of real numbers). In particular, we study the case where the expansion of $γ+1$ is periodic. Under certain assumptions, the base satisfies algebraic properties. We compute the bounds for the norms of the nontrivial Galois conjugates associated with the base; thereby, extending the results of Solomyak on the classical beta expansions. |
| title | An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases |
| topic | Number Theory 11A63, 11R09 |
| url | https://arxiv.org/abs/2505.03166 |