论文标题
有效的ERDőS-Wintner定理用于数字扩展
Effective Erdős-Wintner theorems for digital expansions
论文作者
论文摘要
在1972年，delange在类比的情况下观察到了$ q $ addive functions $ f（n）$具有分布函数，并且仅当两个系列$ \ sum f（d q^j）$，$ \ sum f（d q^j）^2 $ commente时，才具有分布功能。本文的目的是提供该定理的定量版本以及对其他类型的数字扩展的概括。除了$ Q $ - AR和CANTOR案例外，我们还专注于基于斐波那契序列的Zeckendorf扩展，在该序列中，我们为存在分配功能提供了足够且必要的条件，即两个$ \ sum f（f_j）$，$ \ sum，$ \ sum f（f_jj）^2 $ formange（以前仅融合了一个条件）。
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).
