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arXiv · 2510.22858

Effective Erd\H{o}s--Wintner for Cantor numeration systems via a trailing-window method

Johann Verwee
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Abstract

We prove explicit Erd\H{o}s--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.

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Johann Verwee. 2025-10-26. Effective Erd\H{o}s--Wintner for Cantor numeration systems via a trailing-window method. https://arxiv.org/abs/2510.22858

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