Effective Erdős--Wintner for Cantor numeration systems via a trailing-window method
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914242166259712 |
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| author | Verwee, Johann |
| author_facet | Verwee, Johann |
| contents | We prove explicit Erdős--Wintner bounds for Cantor numeration systems via a simple trailing-window decomposition. We temporarily discard the last block of digits (the ``window'') and analyze the remaining prefix. The resulting bound has three contributions: (i) a bridge loss from discarding the window; (ii) a variance-type tail for the prefix; and (iii) a regime-dependent smoothing term (Esseen, bounded density, or cancellation of the third cumulant). Optimizing the window length yields rates that are explicit in the sample size. In the fixed-base (q-adic) case we recover Delange's product and obtain effective convergence bounds; the same scheme applies unchanged to Cantor numeration systems. We also include a brief guide indicating when each regime is preferable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22858 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective Erdős--Wintner for Cantor numeration systems via a trailing-window method Verwee, Johann Number Theory Probability 11K06 (Primary) 11N37, 11K38, 11A63, 60F05 (Secondary) We prove explicit Erdős--Wintner bounds for Cantor numeration systems via a simple trailing-window decomposition. We temporarily discard the last block of digits (the ``window'') and analyze the remaining prefix. The resulting bound has three contributions: (i) a bridge loss from discarding the window; (ii) a variance-type tail for the prefix; and (iii) a regime-dependent smoothing term (Esseen, bounded density, or cancellation of the third cumulant). Optimizing the window length yields rates that are explicit in the sample size. In the fixed-base (q-adic) case we recover Delange's product and obtain effective convergence bounds; the same scheme applies unchanged to Cantor numeration systems. We also include a brief guide indicating when each regime is preferable. |
| title | Effective Erdős--Wintner for Cantor numeration systems via a trailing-window method |
| topic | Number Theory Probability 11K06 (Primary) 11N37, 11K38, 11A63, 60F05 (Secondary) |
| url | https://arxiv.org/abs/2510.22858 |