On the intersections of localized Jarník sets and localized uniformly Jarník sets in continued fractions (Q1984245)

Let \(a_{n}(x)\) be the \(n\)-th digit of the continued fraction expansion of \(x\in[0,1)\) and let \(q_{n}(x)\) be the denominator of the \(n\)-the rational convergent of \(x\). The authors are interested in localized Jarnik sets \[ J=\{x\in[0,1)|\liminf_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_{n}(x)}=\alpha(x), \limsup_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_{n}(x)}=\beta(x)\}\] for non-negative continuous functions \(\alpha,\beta\) on \([0,1]\) with \(\alpha(x)\le \beta(x)\). They prove that the Hausdorff dimension of \(J\) is given by \(1/(\min \beta+2)\) if \(\min \alpha>0\). In addition they prove that the set \(\{\dim_{H}J|\alpha,\beta\mbox{ with }\min \alpha=0\}\) is dense in \([0,1]\). The results are interesting generalizations of \textit{B. Tan} and \textit{Q. Zhou} [J. Math. Anal. Appl. 478, No. 1, 229--235 (2019; Zbl 1417.37058)], where \(\alpha\) and \(\beta\) are constant.