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Effective Erdős-Wintner for Cantor series via a trailing-window method

Johann Verwee

TL;DR

The paper develops an explicit, trailing-window Erdős–Wintner framework for Cantor series that separates a trailing window from a prefix. It yields a universal bound combining a bridge loss, a variance-type prefix tail, and a regime-dependent smoothing term, with three regimes (Esseen baseline, bounded density, and third-cumulant cancellation) that can be chosen based on available structure. By optimizing the window length, it provides explicit rates in $N$ and recovers known results in the $q$-adic setting (Delange product) while remaining applicable to general Cantor series. The approach offers a modular toolkit for deriving effective distributional limits and can yield sharper rates when additional regularity or symmetry is present in the digit map. This has practical impact for quantitative distributional approximations of digit-based additive functions across Cantor-type bases.

Abstract

We prove explicit Erdős--Wintner bounds for Cantor series 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 series. We also include a brief guide indicating when each regime is preferable.

Effective Erdős-Wintner for Cantor series via a trailing-window method

TL;DR

The paper develops an explicit, trailing-window Erdős–Wintner framework for Cantor series that separates a trailing window from a prefix. It yields a universal bound combining a bridge loss, a variance-type prefix tail, and a regime-dependent smoothing term, with three regimes (Esseen baseline, bounded density, and third-cumulant cancellation) that can be chosen based on available structure. By optimizing the window length, it provides explicit rates in and recovers known results in the -adic setting (Delange product) while remaining applicable to general Cantor series. The approach offers a modular toolkit for deriving effective distributional limits and can yield sharper rates when additional regularity or symmetry is present in the digit map. This has practical impact for quantitative distributional approximations of digit-based additive functions across Cantor-type bases.

Abstract

We prove explicit Erdős--Wintner bounds for Cantor series 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 series. We also include a brief guide indicating when each regime is preferable.
Paper Structure (19 sections, 10 theorems, 95 equations)

This paper contains 19 sections, 10 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. A general Erdős--Wintner criterion for Cantor series
  3. Effective Erdős--Wintner for Cantor series
  4. Method selection (Cantor case).
  5. Remark (the $q$--adic case).
  6. Final forms in $N$.
  7. (B) dominates (A) and (C).
  8. (C) dominates (A) when density is unknown.
  9. (A) dominates when neither density nor cancellation is available.
  10. (I) Polynomial tails on the digits.
  11. (II) Geometric tails: Cantor--Lebesgue type.
  12. Van der Corput in Cantor base.
  13. On the optimal $(\log N)/N$ rate.
  14. Particular case: q-adic
  15. Delange’s $q$-adic Erdős--Wintner product.
  16. ...and 4 more sections

Key Result

Theorem 2.1

Assume $f$ is $Q$-additive. Then there exists a cumulative distribution function (c.d.f.) $F$, that is if and only if In that case, the characteristic function of $F$ is

Theorems & Definitions (16)

  • Theorem 2.1: Erdős--Wintner for $Q$-additive functions in constant-like Cantor series
  • proof : Proof of Theorem \ref{['th:Erdős--Wintner-Cantor']}
  • Theorem 3.1: Unified Cantor window bound
  • proof
  • Lemma 3.2: Esseen smoothing, one–sided form
  • Corollary 3.3: Regime (A): baseline Esseen
  • Corollary 3.4: Regime (B): bounded density
  • Corollary 3.5: Regime (C): third cumulant cancels
  • Theorem A.1: Zolotarev zeta3 route
  • proof : Sketch
  • ...and 6 more