A generalization of the Jarník-Besicovitch theorem by continued fractions (Q2813950)

This article deals with the following generalization of the \textit{Jarník-Besicovitch theorem} by continued fractions.NEWLINENEWLINESome notations and definitions. {\parindent=0.7cm \begin{itemize}\item[--] The authors use the notation \(|I|\) for the length of an interval \(I\), the abbreviation `i.o.' for \textit{infinitely often} and \(\dim_H\) for \textit{Hausdorff dimension}. \item[--] Continued fractions expansions can be induced by the Gauss transformation \(T\) defined by \(T: [0,1)\rightarrow [0,1)\) given by \(T(0):=0\), \(T(x)=1/x \pmod 1\) for \(x\in(0,1).\) Each irrational number \(x\in[0,1)\) admits a unique infinite continued fraction expansion \(x=\frac{1}{a_1(x)+\frac{1}{a_2(x)+\dots}}\) where \(a_1(x)=\lfloor x^{-1}\rfloor\) , \(a_n(x)=a_1(T^{n-1}(x))\in\mathbb N\) for all \(n\geq 1\) called \textit{partial quotients } of \(x\). \item[--] \(S_n g(x)=g(x)+\dots+g(T^{n-1} x)\) denotes the \textit{ergodic sum} of \(g\). \item[--] Let \(\tau(x), f(x)\) be two positive functions defined on \([0,1]\). A point \(x\in[0,1)\) is saidNEWLINENEWLINE(1)\textit{\(f\)-approximable} if \(a_{n+1}(x)\geq e^{S_nf(x)}\), i.o. \(n\in\mathbb N\);NEWLINENEWLINE(2) \textit{localized \((\tau; f)\)-approximable} if \(a_{n+1}(x)\geq e^{\tau(x)\cdot S_n f(x)}\), i.o. \(n\in\mathbb N,\)NEWLINENEWLINE(3) \textit{exactly localized \((\tau ; f)\)-approximable} if for any \(\varepsilon>0, a_{n+1}(x)\geq e^{(\tau(x)-\varepsilon)\cdot S_n f(x)}\), i.o. but \(a_{n+1}(x)\leq e^{(\tau(x)+\varepsilon)S_nf(x)}\) eventually. The collection of points defined above denoted \(\mathrm{Diop}^{(i)} (\tau; f), (i=1,2,3)\) are called the \textit{generalized Jarník-Besicovitch sets}. \item[--] For any \(n\geq 1\) and \((a_1,a_2,\dots,a_n)\in\mathbb N^n\) , the set \(I_n(a_1,a_2,\dots,a_n)=\{x\in[0,1): a_1(x)=a_1,\dots,a_n(x)=a_n\}\) called an \textit{\(n\)th-order cylinder,} is the collections of points whose expansions begins with \((a_1,\dots, a_n)\). \item[--] The \(n\)th variation of \(f\) is defined as Var\(_n(f):=\sup_{x,y: I_n(x)=I_n(y)}|f(x)-f(y)|\), where \(I_n(x)\) is the \(n\)th order cylinder containing \(x\) in the continued fraction expansion. \item[--] A function \(f\) is said to fulfill the \textit{tempered distortion property} if \(\mathrm{Var}_1(f)<\infty\) and \(\mathrm{Var}_n(f)\rightarrow 0\) as \(n\rightarrow\infty\). \item[--] Let \(\Psi\) be a function with the tempered distortion property. The \textit{pressure function} \(P(T,\psi)\) with respect to the \textit{potential} \(\psi\) and the Gauss system \(([0,1),T)\) is defined by NEWLINE\[NEWLINEP(T,\psi)=\lim_{n\rightarrow\infty}\frac{1}{n}\log \sum _{a_1,\dots,a_n\in \mathbb N}\sup_{x\in[0,1)} e^{S_n\psi([a_1,\dots, a_n+x])}.NEWLINE\]NEWLINE For details on the pressure function see [\textit{P. Walters}, An introduction to ergodic theory. New York-Heidelberg-Berlin: Springer-Verlag (1982; Zbl 0475.28009)]. NEWLINENEWLINE\end{itemize}} The main result:NEWLINENEWLINETheorem. Let \(\tau: [0,1]\rightarrow \mathbb R_{\geq 0}\) be a non-negative continuous function and \(f: [0,1]\rightarrow \mathbb R^+\) satisfy the tempered distortion property. Then the Hausdorff dimensions of \(\mathrm{Diop}^{(i)}(\tau;f) (i=1,2,3)\) are all given as \(\inf\{s\geq 0 : P(T,-s(\tau_{\min} f+\log|T'|))\leq 0\}\) where \(\tau_{\min}=\min\{\tau(x) : x\in [0,1]\}\).NEWLINENEWLINESee also [\textit{Y. Bugeaud}, Math. Ann. 327, No. 1, 171--190 (2003; Zbl 1044.11059); Unif. Distrib. Theory 3, No. 2, 9--20 (2008; Zbl 1212.11071); \textit{Y. Bugeaud} and \textit{C. G. Moreira}, Acta Arith. 146, No. 2, 177--193 (2011; Zbl 1211.11084)] for the Hausdorff dimension of the \textit{sets of points with exact approximation order}.