Finite pattern problems related to Engel expansion

Let $\mathcal{F}$ be a countable collection of functions $f$ defined on the integers with integer values, such that for every $f\in \mathcal{F}$, $f(n)\to +\mathrm{\infty }$ as $n\to +\mathrm{\infty }$. This paper primarily investigates the Hausdorff dimension of the set of points whose digit sequences of the Engel expansion are strictly increasing and contain any finite pattern of $\mathcal{F}$, demonstrating applications with representative examples.