Metric theorems for continued $\beta$-fractions

Abstract

Let $\beta >1$ be a root of the polynomial $t^2=a t+1$ with $a\in {\mathbb {N}}, a\ge 1$ or a root of the polynomial $t^2=a t-1$ with $a\in {\mathbb {N}}, a\ge 3$. In this paper, we consider the metric properties of the continued $\beta$-fractions. We show that the Lebesgue measure of the following set

$$\begin{aligned} E(\varphi )=\big \{x\in [0,1): a_n(x)\ge \varphi (n)\; \text {for infinitely many}\;n\in {\mathbb {N}}\big \} \end{aligned}$$

is null or full according to the convergence or divergence of the series $\sum _{n=1}^{\infty }\frac{1}{\varphi (n)}$, where $a_n(x)$ is the n-th partial quotients in the continued $\beta$-fraction expansion of x and $\varphi$ is a postive function defined on ${\mathbb {N}}$. As a result, the set of numbers in the interval [0, 1) with bounded partial quotients in their continued $\beta$-fraction expansions is of zero Lebesgue measure.