Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion

Abstract:

Let $Q=\{q_k\}_{k\geq 1}$ be a sequence of positive integers with $q_k\geq 2$ for every $k\geq 1$. Then each point $x\in [0,1]$ is attached with an infinite series expansion \[ x=\frac {\varepsilon _1(x)}{q_1}+\frac {\varepsilon _2(x)}{q_1q_2}+\cdots +\frac {\varepsilon _n(x)}{q_1\cdots q_n}+\cdots , \] which is called the Cantor series expansion of $x$. In this paper, we study the shrinking target problems for the system induced by the Cantor series expansion. More precisely, put $T_{Q}^{n}(x)=q_1\cdots q_nx-\lfloor q_1\cdots q_nx\rfloor$; the shrinking target problem in such a nonautonomous system can be formulated as considering the size of the set \[ \mathbb {E}_y(\varphi ):=\{x\in [0,1]:~ |T_{Q}^{n}(x)-y|<\varphi (n) \text { i. o. }n\},\] where $y$ is a fixed point in $[0,1]$ and $\varphi : \mathbb {N}\to (0,1)$ is a positive function with $\varphi (n)\to 0$ as $n\to \infty$. It is proved that both the Lebesgue measure and the Hausdorff measure of $\mathbb {E}_{y}(\varphi )$ fulfill a dichotomy law according to the divergence or convergence of certain series.