Metric theory for continued \(\beta \)-fractions (Q6627033)

Suppose \(\beta >1\) is a real number; then an expansion of a number \(x\) of the following form \N\[\N x=\sum^{\infty} _{n=1}{\frac{\varepsilon_n(x, \beta)}{\beta^n}} := ( \varepsilon_1(x,\beta), \varepsilon_2(x, \beta),\dots , \varepsilon_n(x, \beta), \dots ), \N\]\Nwhere \(\varepsilon_n(x, \beta) \in \{0, 1, \dots , [\beta]\}\) when \(\beta \notin \mathbb N\) and \(\varepsilon_n(x, \beta) \in \{0, 1, \dots , \beta -1\}\) when \(\beta \in \mathbb N\), is \emph{a \(\beta\)-expansion}, that generated by the \(\beta\)-transformation \(T_\beta: [0, 1) \to [0, 1)\) of the form \N\[\NT_\beta (x) =\beta x- [\beta x], \N\]\Nwhere \([\beta x]\) denotes the largest integer no more than \(\beta x\).\N\NThe continued \(\beta\)-fraction expansion can be generated by the map \(T : [0, 1) \to [0, 1)\) given by \(T(0)=0\) and \N\[\NT(x) = \left\{\frac{1}{x}\right\}_\beta = \frac{1}{x} -\left[\frac{1}{x}\right]_\beta \N\]\Nfor \(x\in (0, 1)\). Here \(\{x\}_\beta\) and \([x]_\beta\) denote the \(\beta\)-fractional and \(\beta\)-integral part of \(x\) respectively. In addition, \N\[\Na_n(x) := \left[\frac{1}{T^{n-1}(x)}\right]_\beta. \N\]\NThe present research deals with the Lebesgue measure and the Hausdorff dimension for some set, which is defined by a certain positive function and the continued \(\beta\)-fraction expansion.\N\NSuppose \( \varphi : \mathbb N \to \mathbb R^+ \) is a positive function, \(\varepsilon^\ast(1, \beta) = (\varepsilon_1^\ast(1, \beta), \varepsilon_2^\ast(1, \beta), \cdots)\) be the infinite \(\beta\)-expansion of \(1\), and \(A_0 = \{\beta > 1 : \{ \ell_n(\beta), n\ge 1 \} ~\text{is bounded} \}\), where \(\ell_n(\beta) := \sup \{j \geq 0 : \varepsilon_n^\ast(1, \beta) = \cdots = \varepsilon_{n + j}^\ast(1, \beta) = 0 \}\) is the maximal length of consecutive zeros in \(\varepsilon^\ast(1, \beta)\) starting from \(\varepsilon_n^\ast(1, \beta)\).\N\NSo, the main attention is given to properties of the following set: \N\[\NE(\varphi) = \{ x \in [0, 1) : a_n (x) \geq \varphi (n) \text{ for infinitely many } n \in \mathbb{N} \}. \N\]\NIt is noted that for any fixed \(\beta \in A_0\), the Lebesgue measure of \(E(\varphi)\) is null or full according to the convergence or divergence of the series \N\[\N\sum_{n = 1}^\infty \frac{ 1}{ \varphi (n)} \N\]\Nrespectively. The numbers \(B\) and \(b\) are parameters for values of the Hausdorff dimensions of the set, where \N\[\N\log B=\liminf_{n\to\infty}{\frac{\log \varphi(n)}{n}} \N\]\N and \N\[\N\log b=\liminf_{n\to\infty}{\frac{\log\log \varphi(n)}{n}}.\N\]\NSpecial attention is also given to explanations of additional notions, to discussions of known related results, and to auxiliary proofs.