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Effective Erdős-Wintner theorems for digital expansions. (English) Zbl 1479.11170

The purpose of this paper is to consider the analogue problem in the context of digital expansions, where it is restricted to the (common) \(q\)-ary digital expansion, to Cantor digital expansions, and to the Zeckendorf expansion. In this paper the focus is on analogues of the Erdős-Wintner theorems for digital expansions and their quantitative versions. For the \(q\)-adic case there is already a proper analogue by H. Delange [Acta Arith. 21, 285–298 (1972; Zbl 0219.10062)] saying that a real-valued \(q\)-additive function \(f(n)\) has a distribution function \(F(y)=\lim_{N\to\infty}(1/N)\#\{n<N: f(n)\le y\}\) if and only if the two series \(\sum_{j\ge 0}\sum_{d=1}^{q-1}f(d q^j)\) and \(\sum_{j\ge 0}\sum_{d=1}^{q-1}f(d q^j)^2\) converge. By applying a Berry-Esseen inequality a quantified version of this theorem is presented and examples are given, the most interesting one being related to Cantor-Lebesgue measures. The Zeckendorf digital expansions are also discussed, where the base sequence are the Fibonacci numbers. A sufficient and necessary condition for the existence of a distribution function is given as that the two series \(\sum_{j\ge 2}f(F_j)\) and \(\sum_{j\ge 2}f(F_j)^2\) converge (so far only a sufficient condition was known [G. Barat and P. J. Grabner, J. Number Theory 60, No. 1, 103–123 (1996; Zbl 0862.11048)]).

MSC:

11N60 Distribution functions associated with additive and positive multiplicative functions
11A67 Other number representations
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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