A Normality Conjecture on Rational Base Number Systems
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918431597527040 |
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| author | Andrieu, Mélodie Eliahou, Shalom Vivion, Léo |
| author_facet | Andrieu, Mélodie Eliahou, Shalom Vivion, Léo |
| contents | The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words. We conjecture that every minimal and maximal word is normal over an appropriate subalphabet. To support this conjecture, we present extensive numerical experiments that examine the richness threshold and the deviation from normality of these words. We also discuss the implications that the validity of our conjecture would have for several long-standing open problems, including the existence of $Z$-numbers (Mahler, 1968) and $Z_{p/q}$-numbers (Flatto, 1992), the existence of triple expansions in rational base $p/q$ (Akiyama, 2008), and the Collatz-inspired `4/3 problem' (Dubickas and Mossinghoff, 2009). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Normality Conjecture on Rational Base Number Systems Andrieu, Mélodie Eliahou, Shalom Vivion, Léo Number Theory Combinatorics 11A67, 68R15 (Primary) 11K16, 11J71 (Secondary) The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words. We conjecture that every minimal and maximal word is normal over an appropriate subalphabet. To support this conjecture, we present extensive numerical experiments that examine the richness threshold and the deviation from normality of these words. We also discuss the implications that the validity of our conjecture would have for several long-standing open problems, including the existence of $Z$-numbers (Mahler, 1968) and $Z_{p/q}$-numbers (Flatto, 1992), the existence of triple expansions in rational base $p/q$ (Akiyama, 2008), and the Collatz-inspired `4/3 problem' (Dubickas and Mossinghoff, 2009). |
| title | A Normality Conjecture on Rational Base Number Systems |
| topic | Number Theory Combinatorics 11A67, 68R15 (Primary) 11K16, 11J71 (Secondary) |
| url | https://arxiv.org/abs/2510.11723 |