Beta-expansion and continued fraction expansion

For any real number β>1β>1, let ε(1,β)=(ε1(1),ε2(1),…,εn(1),…)ε(1,β)=(ε1(1),ε2(1),…,εn(1),…) be the infinite β  -expansion of 1. Define ln=sup{k⩾0:εn+j(1)=0for all1⩽j⩽k}. Let x∈[0,1)x∈[0,1) be an irrational number. We 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  . If {ln,n⩾1} is bounded, we obtain that for all x∈[0,1)∖Qx∈[0,1)∖Q,lim infn→+∞kn(x)n=logβ2β*(x),lim supn→+∞kn(x)n=logβ2β*(x), where β*(x)β*(x), β*(x)β*(x) are the upper and lower Lévy constants, which generalize the result in [J. Wu, Continued fraction and decimal expansions of an irrational number, Adv. Math. 206 (2) (2006) 684–694]. Moreover, if lim supn→+∞lnn=0, we also get the similar result except a small set.