Exact approximation order of real numbers in Cantor series expansions

Let $Q=\{q_n{\}}_{n\ge 1}$ be a sequence of integers with $q_n\ge 2$ for all $n\in \mathbb{N}$. For any real number $x\in [0,1)$, it can be expanded into the following infinite series: $$x=\frac{{\epsilon }_1(x)}{q_1}+\frac{{\epsilon }_2(x)}{q_1{q}_2}+\cdots +\frac{{\epsilon }_n(x)}{q_1{q}_2\cdots q_n}+\cdots ,$$ which is called the Cantor series expansion of $x$. We introduce the exact spproximation order in Cantor series expansions. It is analogous to the notion appearing in classical Diophantine approximation. More precisely, let ${\omega }_n(x)$ denote the $n$-th partial sum of the Cantor series expansion of $x$. For any monotonic function $\psi$, we study the metric theory of the set $E_c(\psi )$ of points that are exactly $\psi$-approximable by ${\omega }_n(x)$.