Exact approximation order of real numbers in Cantor series expansions

Abstract

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: $$x =\frac{\varepsilon_1(x)}{q_1}+ \frac{\varepsilon_2(x)}{q_1 q_2}+ \cdots+ \frac{\varepsilon_n(x)}{q_1 q_2 \cdots q_n}+ \cdots,$$ 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 $\omega_n(x)$.