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 every $n\ge 1$. Every $x\in [0,1)$ admits a Cantor series expansion of the form $$x=\frac{{\epsilon }_1(x)}{q_1}+\frac{{\epsilon }_2(x)}{q_1{q}_2}+\cdots +\frac{{\epsilon }_n(x)}{q_1{q}_2\cdots q_n}+\cdots .$$In this paper, we study exact approximation of real numbers by the $n$-th partial sums of their Cantor series expansions. More precisely, for a non-increasing function $\psi :\mathbb{N}\to (0,\infty )$ satisfying $\psi (n)\to 0$, let $E_c(\psi )$ be the set of points that are $\psi$-approximable by their $n$-th partial sums but are not $b\psi$-approximable for any $0<b<1$. We determine the Hausdorff dimension of $E_c(\psi )$.