Beta-expansion and continued fraction expansion of real numbers
Volume 187 / 2019
Abstract
Let $\beta \gt 1$ 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 $\beta$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\log\beta)/\pi^2$ 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 $(6\log2\log\beta)/\pi^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre (1997) from $\beta = 10$ to any $\beta \gt 1$. Moreover, we also discuss whether $\beta$-expansion or continued fraction expansion yields better approximations of real numbers.
