On finite $β$-expansions for the set of natural numbers
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908468191952896 |
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| author | Carvalho, Túlio O. Moreira, Catharina M. |
| author_facet | Carvalho, Túlio O. Moreira, Catharina M. |
| contents | We present a study of the problem of finiteness of the $β$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $β$-expansion of 1 is not a non-decreasing sequence. We show a class of simple $β$-numbers which display a puzzling behaviour, in the sense that an infinite subset of such $β$ satisfy $F_1$, whereas a complementary infinite subset does not. This puzzle is organized by the residue class modulo 3 of an integer coefficient of the main family of polynomials discussed. We prove that, when $F_1$ does not hold, the complement of ${{\rm Fin}(β)}$ is infinite. Finally, we give a concise sufficient condition for $F_1$, which is the finitude of the $β$-expansion of $(β-\lfloor β\rfloor)^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On finite $β$-expansions for the set of natural numbers Carvalho, Túlio O. Moreira, Catharina M. Number Theory Dynamical Systems 11K16, 37E15, 11A63 We present a study of the problem of finiteness of the $β$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $β$-expansion of 1 is not a non-decreasing sequence. We show a class of simple $β$-numbers which display a puzzling behaviour, in the sense that an infinite subset of such $β$ satisfy $F_1$, whereas a complementary infinite subset does not. This puzzle is organized by the residue class modulo 3 of an integer coefficient of the main family of polynomials discussed. We prove that, when $F_1$ does not hold, the complement of ${{\rm Fin}(β)}$ is infinite. Finally, we give a concise sufficient condition for $F_1$, which is the finitude of the $β$-expansion of $(β-\lfloor β\rfloor)^2$. |
| title | On finite $β$-expansions for the set of natural numbers |
| topic | Number Theory Dynamical Systems 11K16, 37E15, 11A63 |
| url | https://arxiv.org/abs/2507.19604 |