Exact approximation order of real numbers in Cantor series expansions

Abstract

Let $Q=\{q_n\}_{n\geq1}$ be a sequence of integers with $q_n\geq2$ for every $n\geq1$. Every $x\in[0,1)$ admits a Cantor series expansion of the form $$ x = \frac{\varepsilon_1(x)}{q_1} + \frac{\varepsilon_2(x)}{q_1q_2} +\cdots+ \frac{\varepsilon_n(x)}{q_1q_2\cdots q_n} +\cdots.$$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 $ψ:\mathbb{N}\to(0,\infty)$ satisfying $ψ(n)\to0$, let $E_c(ψ)$ be the set of points that are $ψ$-approximable by their $n$-th partial sums but are not $bψ$-approximable for any $0<b<1$. We determine the Hausdorff dimension of $E_c(ψ)$.