Continued fraction and decimal expansions of an irrational number

For an irrational number x   and n⩾1n⩾1, 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 decimals of x   and pn(x)/qn(x)pn(x)/qn(x) the nth convergent of x. Letβ∗(x)=lim infn→∞logqn(x)n,β∗(x)=lim supn→∞logqn(x)n. We prove thatlim supn→∞kn(x)n=log102β∗(x),lim infn→∞kn(x)n=log102β∗(x). This result significantly strengthens the results of G. Lochs and C. Faivre.