Effective Erdos-Wintner theorems for digital expansions

[摘要] In 1972 Delange [9] observed in analogy of the classical Erdos-Wintner theorem that q -additive functions f (n) has a distribution function if and only if the two series Sigma f(dqj), Sigma f (dq(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 qary 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 Sigma f(F-j), Sigma f(F-j)(2) converge (previously only a sufficient condition was known [2]). (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Erdos-Wintner theorem;Digital expansions