Erdős-Wintner theorem for linear recurrent bases
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914271795871744 |
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| author | Verwee, Johann |
| author_facet | Verwee, Johann |
| contents | Let $(G_n)_{n\geqslant 0}$ be a linear recurrence sequence defining a numeration system and satisfying mild structural hypotheses. For real-valued G-additive functions (additive in the greedy G-digits), we establish an Erdős-Wintner-type theorem: convergence of two canonical series (a first-moment series and a quadratic digit-energy series) is necessary and sufficient for the existence of a limiting distribution along initial segments of the integers. In that case, the limiting characteristic function admits an explicit infinite-product factorization whose local factors depend only on the underlying digit system. We also indicate conditional extensions of this two-series criterion to Ostrowski numeration systems with bounded partial quotients and to Parry $β$-expansions with Pisot-Vijayaraghavan base $β$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Erdős-Wintner theorem for linear recurrent bases Verwee, Johann Number Theory 11N60, 11A63, 11K16 Let $(G_n)_{n\geqslant 0}$ be a linear recurrence sequence defining a numeration system and satisfying mild structural hypotheses. For real-valued G-additive functions (additive in the greedy G-digits), we establish an Erdős-Wintner-type theorem: convergence of two canonical series (a first-moment series and a quadratic digit-energy series) is necessary and sufficient for the existence of a limiting distribution along initial segments of the integers. In that case, the limiting characteristic function admits an explicit infinite-product factorization whose local factors depend only on the underlying digit system. We also indicate conditional extensions of this two-series criterion to Ostrowski numeration systems with bounded partial quotients and to Parry $β$-expansions with Pisot-Vijayaraghavan base $β$. |
| title | Erdős-Wintner theorem for linear recurrent bases |
| topic | Number Theory 11N60, 11A63, 11K16 |
| url | https://arxiv.org/abs/2512.20882 |