Erd\H{o}s-Wintner theorem for linear recurrent bases

TL;DR

This paper extends the Erdős-Wintner theorem to real-valued additive functions defined on linear recurrence-based numeration systems, providing criteria for the existence of limiting distributions and explicit characteristic functions.

Contribution

It establishes an Erdős-Wintner-type criterion for linear recurrence numeration systems and derives explicit formulas for limiting distributions, extending to certain Ostrowski and Parry expansions.

Findings

Convergence of two series characterizes limiting distributions.

Explicit infinite-product form of the characteristic function is derived.

Extensions to Ostrowski and Parry numeration systems are discussed.

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 Erd\H{o}s-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 $\beta$-expansions with Pisot-Vijayaraghavan base $\beta$.