Effective Erd\H{o}s-Wintner theorems for digital expansions

TL;DR

This paper extends Erd ext{"o}s-Wintner theorems to digital expansions like $q$-ary, Cantor, and Zeckendorf, providing quantitative criteria for the existence of distribution functions of additive functions.

Contribution

It offers quantitative versions and necessary and sufficient conditions for distribution functions in various digital expansions, including the Zeckendorf expansion.

Findings

Convergence of series $ ext{sum} f(d q^j)$ and $ ext{sum} f(d q^j)^2$ characterizes distribution functions for $q$-additive functions.

Necessary and sufficient conditions established for Zeckendorf expansion based on Fibonacci numbers.

Generalizations to multiple digital expansion systems beyond classical cases.

Abstract

In 1972 Delange observed in analogy of the classical Erd\H os-Wintner theorem that $q$-additive functions $f(n)$ has a distribution function if and only if the two series $\sum f(d q^j)$, $\sum f(d q^j)^2$ converge. The purpose of this paper is to provide quantitative versions of this theorem as well as generalizations to other kinds of digital expansions. In addition to the $q$-ary and Cantor case we focus on the Zeckendorf expansion that is based on the Fibonacci sequence, where we provide a sufficient and necessary condition for the existence of a distribution function, namely that the two series $\sum f(F_j)$, $\sum f(F_j)^2$ converge (previously only a sufficient condition was known).