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Beta-expansion and continued fraction expansion of real numbers

Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\logβ)/π^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\logβ)/π^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $β= 10$ to any $β>1$. Moreover, we also discuss which of the $β$-expansion and continued fraction expansion yields the better approximations of real numbers.