Let \(\beta>1 \) be a real number and \(T_\beta: [0,1) \to [0,) \) be a the \(\beta\)-transformation defined as \(T_\beta(x)=\beta x-[\beta x]. \) Every \(x \in [0,1) \) can be uniquely expanded into a finite or infinite series, i.e., \[ x=\frac{\varepsilon_1(x)}{\beta}+\frac{\varepsilon_2(x)}{\beta^2}+\dots+\frac{\varepsilon_n(x)}{\beta^n}+\dots, \] where \(\varepsilon_1(x)=[\beta x]\) and \(\varepsilon_{n+1}(x)=\varepsilon_1\big(T^n_\beta x\big)\) for all \(n\geq 1 \). This representation is called the \(\beta\)-expansion of \(x\) and denoted \((\varepsilon_1(x),\varepsilon_2(x),\dots)\).
Let \(T:[0,1) \to [0,1) \) be the Gauss transformation given by \(T(0):=0, T(x):=1/x -[1/x] \) if \(x \in (0,1).\) Any irrational number \(x\in [0,1) \) can be written as \[ x=\cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\genfrac{}{}{0pt}{0}{}{\ddots + \cfrac{1}{a_n(x)+\genfrac{}{}{0pt}{0}{}{\ddots }}}}}, \] where \(a_1(x)=[1/x] \) and \(a_{n+1}(x)=a_1\big(T^n x \big) \) for all \(n \geq 1. \) This representation is called the continued fraction expansion of \(x\) and denoted \([a_1(x),a_2(x),\dots]\). Let \(J(\varepsilon_1(x),\dots,\varepsilon_n(x))\) and \(I(a_1(x),\dots,a_m(x))\) be the cylinders of the \(\beta\)-expansion and the continued fraction expansion respectively and \[ k_n(x)=\sup \{m\geq 0: J(\varepsilon_1(x),\dots,\varepsilon_n(x)) \subseteq I(a_1(x),\dots,a_m(x)) \} \] The following theorem is proved in this article.
Theorem. Let \(\beta>1 \) be a real number. For any \(\varepsilon >0, \) there exist positive \(A\) and \(\alpha \) (both depending on \(\beta\) and \(\alpha \) ) such that for all \( n\geq 1\), \[ \lambda \bigg\{ x\in [0,1) : \bigg|\frac{k_n(x)}{n}-\frac{6\log 2 \log \beta }{\pi^2} \bigg|\geq \varepsilon \bigg\} \leq A e^{-\alpha n}, \] \(\lambda\) is the Lebesgue measure.