Beta-expansion and continued fraction expansion of real numbers

Abstract

Let $\beta >1$ 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 $(6\log 2\log \beta )/{\pi }^2$ 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 $(6\log 2\log \beta )/{\pi }^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $\beta =10$ to any $\beta >1$. Moreover, we also discuss which of the $\beta$-expansion and continued fraction expansion yields the better approximations of real numbers.

16 pages. Any comments are welcome. Thank you very much! arXiv admin note: text overlap with arXiv:1601.02202