Effective Erdös-Wintner theorems for digital expansions

Michael Drmota (1) , Johann Verwee (2, 1)

1 TU Wien - Vienna University of Technology = Technische Universität Wien
2 IECL - Institut Élie Cartan de Lorraine

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).

Mots clés: Erdös-Wintner theorem; digital expansions

Domaines: Théorie des nombres [math.NT]

Michael Drmota, Johann Verwee. Effective Erdös-Wintner theorems for digital expansions. 2021. ⟨hal-03189795⟩