Doubly metric theory and simultaneous shrinking target problem in Cantor series expansion

Abstract

Let [Formula: see text] be a sequence of positive integers with [Formula: see text] for every [Formula: see text] and [Formula: see text] be the transformation with [Formula: see text] which induces the Cantor series expansion. In the paper, we show that the Lebesgue measure and generalized Hausdorff measure of the following shrinking target set [Formula: see text] satisfy a dichotomy law, which depends on the convergence or divergence of a certain series, where [Formula: see text] is a positive function such that [Formula: see text] as [Formula: see text] and i.m. stands for infinitely many. In addition, for every integer [Formula: see text] and integer [Formula: see text] with [Formula: see text], let [Formula: see text] be a Lipschitz function with Lipschitz constant [Formula: see text]. Assume that [Formula: see text] with [Formula: see text], we obtain the Hausdorff dimension of the following set: [Formula: see text] [Formula: see text]