Parry condition, existence and uniqueness of alternate bases
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918395496103936 |
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| author | Charlier, Émilie Kreczman, Savinien Masáková, Zuzana Pelantová, Edita |
| author_facet | Charlier, Émilie Kreczman, Savinien Masáková, Zuzana Pelantová, Edita |
| contents | Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17819 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parry condition, existence and uniqueness of alternate bases Charlier, Émilie Kreczman, Savinien Masáková, Zuzana Pelantová, Edita Number Theory Discrete Mathematics Combinatorics 11A67, 15B48, 11C20, 37B10, 11K16 Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers. |
| title | Parry condition, existence and uniqueness of alternate bases |
| topic | Number Theory Discrete Mathematics Combinatorics 11A67, 15B48, 11C20, 37B10, 11K16 |
| url | https://arxiv.org/abs/2603.17819 |