Convergence exponent of Dirichlet non-improvable numbers in the theory of continued fractions

Xiaoyan Tan, Zhenliang Zhang

Let $x\in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x),\cdots ,a_n(x),\cdots ]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We est…