Multifractal analysis of convergence exponents for products of consecutive partial quotients in continued fractions

For each real number x ∈ (0, 1), let [a₁ (x), a₂ (x), ⋯, a_n (x), ⋯] denote its continued fraction expansion. We study the convergence exponent defined by \(\tau (x): = \inf \left\{{s \ge 0:\sum\limits_{n = 1}^\infty {{{\left({{a_n}(x){a_{n + 1}}(x)} \right)}^{- s}} < \infty}} \right\},\) $$\tau (x):=\inf \left\{s\ge 0:{\displaystyle \sum _{n=1}^{\infty }{\left(a_n(x)a_{n+1}(x)\right)}^{-s}<\infty }\right\},$$ which reflects the growth rate of the product of two consecutive partial quotients. As a main result, the Hausdorff dimensions of the level sets of τ(x) are determined.