Irrationality Exponents and Partial Quotient Growth in Continued Fractions

Let $[a_1(x),a_2(x),\dots ,a_n(x),\dots ]$ be the continued fraction expansion of irrational $x\in [0,1)$, and let $q_n(x)$ be the denominator of the $n$-th convergent. In this paper, we study how the growth rate of $a_{n+1}(x)$ on a prescribed logarithmic size interacts with its upper growth rate relative to $q_n(x)$. For $\psi :\mathbb{N}\to {\mathbb{R}}_{\ge 0}$ satisfying $\psi (n)\to \mathrm{\infty }$ and $\alpha ,\beta \in [0,\mathrm{\infty }]$, define the joint level set $$F_{\alpha ,\beta }:=\{x\in [0,1)\text{\hspace{0.17em}⁣}:\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\log (a_{n+1}(x))}{\psi (n)}=\alpha ,\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\log (a_{n+1}(x))}{\log q_n(x)}=\beta \}\mathrm{.}$$ We determine the Hausdorff dimension of $F_{\alpha ,\beta }$ for all values of $\alpha$ and $\beta$. Our results is related to several earlier results on the metric theory of continued fractions, including those of Bugeaud [Math. Ann. {327} (2003)] on irrationality exponents, Wang–Wu [Adv. Math. {218} (2008)] on the growth of partial quotients, and Song–Tan–Zhang [Nonlinearity {37} (2024)] on the joint distribution of convergence and irrationality exponents.