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arXiv · 2009.05435

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

Michael Drmota·Johann Verwee
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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).

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Michael Drmota, Johann Verwee. 2020-09-11. Effective Erd\H{o}s-Wintner theorems for digital expansions. https://arxiv.org/abs/2009.05435

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