Inhomogeneous and simultaneous Diophantine approximation in Cantor series expansions

Abstract

Let $Q={\left\{q_k\right\}}_{k\ge 1}$ be a sequence of positive integers with $q_k\ge 2$ for all $k\ge 1$. Then every $x\in [0,1]$ is attached with a Cantor series expansion of the form$x=\frac{{ϵ}_1\left(x\right)}{q_1}+\frac{{ϵ}_2\left(x\right)}{q_1{q}_2}+\cdots +\frac{{ϵ}_n\left(x\right)}{q_1\cdots q_n}+\cdots .$ We study inhomogeneous and simultaneous Diophantine approximation in Cantor series expansions. Several versions of generalized shrinking target sets are defined in our framework. We will give a complete metric theory of these object sets in the sense of Lebesgue measure, Hausdorff measure and Hausdorff dimension.