The main purpose of this paper is to discuss the asymptotic behaviour of the difference sq,k (P(n)) - k(q - 1)/2 where sq,k(n) denotes the sum of the first k digits in the q-ary digital expansion of n and P(x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P, a Strassen type version of the law of the iterated logarithm. Furthermore, we extend these results to the joint distribution of q1-ary and q2-ary digital expansions where q1 and q2 are coprime.
- Functional limit theorem
- Q-ary digital expansion
- Strassen's law of the iterated logarithm
- Sum-of-digits function