Limit theorems related to beta-expansion and continued fraction expansion

Let β>1β>1 be a real number and x∈[0,1)x∈[0,1) be an irrational number. Denote by kn(x)kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x   (n∈Nn∈N). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence {kn,n≥1}{kn,n≥1}, which generalize the results of Faivre [8] and Wu [31] respectively from β=10β=10 to any β>1β>1.