Abstract
The main purpose of this paper is to discuss the asymptotic behaviour of the difference s q,k(P(n)) - k(q-1)/2 where s q,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 q 1-ary and q 2-ary digital expansions where q 1 and q 2 are coprime.
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Drmota, M., Fuchs, M. & Manstavičius, E. Functional limit theorems for digital expansions. Acta Mathematica Hungarica 98, 175–201 (2002). https://doi.org/10.1023/A:1022869708089
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DOI: https://doi.org/10.1023/A:1022869708089