MULTIPLICATIVE DIOPHANTINE APPROXIMATION OF THE EXPANSIONS UNDER DIFFERENT BASES ON A LINE

Abstract

In this paper, we discuss a problem of multiplicative Diophantine approximation of the expansions under different bases restricted on a line. More specifically, let [Formula: see text] be a positive function. A dichotomy law of the Hausdorff measure for the following set: [Formula: see text] is obtained, which depends the convergence or divergence of certain series.