Effective Erd\H{o}s-Wintner theorems for digital expansions
| Autor: | Drmota, Michael, Verwee, Johann |
|---|---|
| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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). Comment: 26 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|