Multifractal analysis of the convergence exponent in continued fractions

Let $x \in [0,1)$ be a real number and denote its continued fraction expansion by $[a_1(x),a_2(x), a_3(x),\cdots]$. The convergence exponent of these partial quotients is defined as $$\tau(x):= \inf\left\{s \geq 0: \sum_{n \geq 1} a^{-s}_n(x)<\infty\right\}.$$ In this paper, we investigate some fundamental properties and multifractal analysis of the exponent $\tau(x)$.