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Erdős-Wintner theorem for linear recurrent bases

Johann Verwee

TL;DR

This work extends the Erdős–Wintner distribution theory to linear recurrence bases (LRB), providing a genuine if-and-only-if criterion for the existence of a limiting distribution of G-additive functions in greedy G-digit systems. The core method is an explicit infinite-product factorization of the limiting characteristic function, driven by two canonical series that encode first-moment drift and digit-energy fluctuations. The authors specialize to order-2 bases (including Zeckendorf's system), establish stability under addition and small digit perturbations, and outline conditional extensions to Ostrowski and beta–PV systems. The results unify and generalize prior EW-type theorems for q-additive and Zeckendorf-type systems while highlighting the role of PV properties and automaton structure in the distributional behavior. They also lay out a program for extensions to broader numeration systems and dynamical hypotheses, with open questions about effective rates and nonstationary settings.

Abstract

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 $β$.

Erdős-Wintner theorem for linear recurrent bases

TL;DR

This work extends the Erdős–Wintner distribution theory to linear recurrence bases (LRB), providing a genuine if-and-only-if criterion for the existence of a limiting distribution of G-additive functions in greedy G-digit systems. The core method is an explicit infinite-product factorization of the limiting characteristic function, driven by two canonical series that encode first-moment drift and digit-energy fluctuations. The authors specialize to order-2 bases (including Zeckendorf's system), establish stability under addition and small digit perturbations, and outline conditional extensions to Ostrowski and beta–PV systems. The results unify and generalize prior EW-type theorems for q-additive and Zeckendorf-type systems while highlighting the role of PV properties and automaton structure in the distributional behavior. They also lay out a program for extensions to broader numeration systems and dynamical hypotheses, with open questions about effective rates and nonstationary settings.

Abstract

Let 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 -expansions with Pisot-Vijayaraghavan base .
Paper Structure (26 sections, 13 theorems, 310 equations)

This paper contains 26 sections, 13 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Linear recurrent sequences and distribution function
  3. Erdős--Wintner theorem for linear recurrent bases
  4. Auxiliary lemmas
  5. Proof of Theorem \ref{['thm:EW-LRB']}
  6. Step 1: Sufficiency.
  7. Step 2 : Necessity.
  8. Examples
  9. LRB of order 2
  10. Stability under addition and small perturbations
  11. Outlook: Ostrowski expansions and beta--PV systems
  12. Local place values and digit bounds.
  13. $\mathcal{S}$--additive functions.
  14. Natural cutoffs.
  15. Two canonical series.
  16. ...and 11 more sections

Key Result

Theorem 3.1

Let $(G_n)_n$ be an LRB and let $f:\mathbb{N}\to\mathbb{R}$ be $G$--additive. Then the following are equivalent: In this case, the limiting characteristic function $\Phi$ admits the infinite product factorization

Theorems & Definitions (19)

  • Definition 2.1: Linear recurrent base (LRB)
  • Theorem 3.1: Erdős--Wintner theorem for LRB
  • Corollary 3.2: Multinacci bases
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Corollary 5.1
  • ...and 9 more