Inhomogeneous and simultaneous Diophantine approximation in Cantor series expansions

Let Q = { q k } k ≥ 1 be a sequence of positive integers with q k ≥ 2 for all k ≥ 1 . Then every x ∈ [ 0 , 1 ] is attached with a Cantor series expansion of the form x = ϵ 1 ( x ) q 1 + ϵ 2 ( x ) q 1 q 2 + ⋯ + ϵ n ( x ) q 1 ⋯ q n + ⋯ . 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.