We investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion.To be precise, let \(Q=\{q_{k}\}_{k\geq1}\) be a sequence of positive integers with \(q_{k}\geq2\) for all \(k\geq1\) . Put \(T_{Q}^{n}(x)=q_{1}\cdots q_{n}x-\lfloor q_{1}\cdots q_{n}x\rfloor\) for each \(n\geq1\) , which gives the \(Q\) -Cantor series expansion.We focus on the following \(\{n_{k},r_{k}\}\) -moving recurrent points proposed by Boshernitzan and Glasner: \(\inf_{k\geq1}|T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|=0,\) where \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) are two given sequences of integers. It is proved that when \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) tend to infinity, the set of \(\{n_{k},r_{k}\}\) -moving recurrent points is of full Lebesgue measure. In addition,we study the size of the following quantitative version of \(\{n_{k},r_{k}\}\) -moving recurrent set: \( R(\{n_{k},r_{k}\}):=\big\{x\in [0,1] : |T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|<\varphi(k)~\text{for i.m.}~k\in \mathbb{N}\big\},\) where \(\varphi \colon \mathbb{N}\rightarrow\mathbb{R}^{+}\) is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) tend to infinity, theLebesgue measure and Hausdorff measure of \(R(\{n_{k},r_{k}\})\) respectively fulfill a dichotomy law according to the convergence or divergence of certain series.
