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 establish, for any \(\alpha ,\beta \) in \([0,+\infty ]\) , the Hausdorff dimension formula of the intersections of the sets of Dirichlet non-improvable numbers and the level set of convergent exponent, i.e. \( G(\alpha ,\beta ): =\left\{ x\in [0,1):\tau (x)=\alpha ,\,\,\text {and} \,\, \limsup _{n\rightarrow \infty }\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}\ge \beta \right\} , \) and \( E(\alpha ,\beta ): =\left\{ x\in [0,1):\tau (x)=\alpha ,\,\,\text {and} \,\, \limsup _{n\rightarrow \infty }\frac{\log (a_n(x)a_{n+1}(x))}{\log q_n(x)}=\beta \right\} , \) where \( \tau (x):= \inf \left\{ s \ge 0: \sum _{n \ge 1} a^{-s}_n(x)<\infty \right\} . \)
