Probabilistic theory of additive functions related to systems of numeration. (English) Zbl 0964.11031

Laurinčikas, A. (ed.) et al., Analytic and probabilistic methods in number theory. Proceedings of the 2nd international conference in honour of J. Kubilius, Palanga, Lithuania, September 23-27, 1996. Utrecht: VSP. New Trends Probab. Stat. 4, 413-429 (1997).
Given a sequence \(b_j\geq 2\) of integers, with \(u_j=b_1...b_j\) we can represent every nonnegative integer uniquely as \(m=\sum d_j u_j\), \(0\leq d_j<b_j\). A function \(f\) is \(U\)-additive, where \(U=(u_j)\), if we have \(f(m)= \sum f(d_ju_j)\). For \(u_j=q^j\) this reduces to the concept of \(q\)-additive functions. These behave similarly to the usual additive functions; often the corresponding results are simpler and are easier to prove.
This paper is a survey with some new results included. The connection with sums of suitably defined independent random variables is systematically explored. The questions considered include moments, concentration, Elliott’s duality, limit laws, models of random processes.
For the entire collection see [Zbl 0896.00021].


11K65 Arithmetic functions in probabilistic number theory
11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.