Numeration systems without a dominant root and regularity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915675759443968 |
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| author | Charlier, Émilie Kreczman, Savinien |
| author_facet | Charlier, Émilie Kreczman, Savinien |
| contents | Positional numeration systems are a large family of numeration systems used to represent natural numbers. Whether the set of all representations forms a regular language or not is one of the most important questions that can be asked of such a system. This question was investigated in a 1998 article by Hollander. Central to his analysis is a property linking positional numeration systems and Rényi numeration systems, which use a real base to represent real numbers. However, this link only exists when the initial numeration system has a dominant root, which is not a necessary condition for regularity. In this article, we show a more general link between positional numeration systems and alternate base numeration systems, a family generalizing Rényi systems. We then take advantage of this link to provide a full characterization of those numeration systems that generate a regular language. We also discuss the effectiveness of our method, and comment Hollander's results and conjecture in the light of ours. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_13180 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numeration systems without a dominant root and regularity Charlier, Émilie Kreczman, Savinien Number Theory Combinatorics 11A67, 68Q45, 11K16, 11A63 Positional numeration systems are a large family of numeration systems used to represent natural numbers. Whether the set of all representations forms a regular language or not is one of the most important questions that can be asked of such a system. This question was investigated in a 1998 article by Hollander. Central to his analysis is a property linking positional numeration systems and Rényi numeration systems, which use a real base to represent real numbers. However, this link only exists when the initial numeration system has a dominant root, which is not a necessary condition for regularity. In this article, we show a more general link between positional numeration systems and alternate base numeration systems, a family generalizing Rényi systems. We then take advantage of this link to provide a full characterization of those numeration systems that generate a regular language. We also discuss the effectiveness of our method, and comment Hollander's results and conjecture in the light of ours. |
| title | Numeration systems without a dominant root and regularity |
| topic | Number Theory Combinatorics 11A67, 68Q45, 11K16, 11A63 |
| url | https://arxiv.org/abs/2512.13180 |