  A fundamental challenge within the metric theory of continued fractions
involves quantifying sets of real numbers, when represented using continued
fractions, exhibit partial quotients that grow at specific rates. For any
positive function $\Phi$, Wang-Wu theorem (2008) comprehensively describes the
Hausdorff dimension of the set \begin{equation*} \EE_1(\Phi):=\left\{x\in [0,
1): a_n(x)\geq \Phi(n) \ {\rm for \ infinitely \ many} \ n\in \N\right\}.
\end{equation*} Various generalisations of this set exist, such as substituting
one partial quotient with the product of consecutive partial quotients in the
aforementioned set which has connections with the improvements to Dirichlet's
theorem, and many other sets of similar nature. Establishing the upper bound of
the Hausdorff dimension of such sets is significantly easier than proving the
lower bound. In this paper, we present a unified approach to get an optimal
lower bound for many known setups, including results by Wang-Wu [Adv. Math.,
2008], Huang-Wu-Xu [Israel J. Math. 2020], Bakhtawar-Bos-Hussain [Nonlinearity
2020], and several others, and also provide a new theorem derived as an
application of our main result. We do this by finding an exact Hausdorff
dimension of the set $$S_m(A_0,\ldots,A_{m-1}) \defeq \left\{ x\in[0,1): \, c_i
A_i^n \le a_{n+i}(x) < 2c_i A_i^n,0 \le i \le m-1 \ \text{for infinitely many }
n\in\N \right\},$$ where each partial quotient grows exponentially and the base
is given by a parameter $A_i>1$. For proper choices of $A_i$'s, this set serves
as a subset for sets under consideration, providing an optimal lower bound of
Hausdorff dimension in all of them. The crux of the proof lies in introducing
of multiple probability measures consistently distributed over the Cantor-type
subset of $S_m(A_0,\ldots,A_{m-1})$.
