Let $F_{n}$ denote the distribution function of the normalized sum of $n$ i.i.d. random variables. In this paper, polynomial rates of approximation of $F_{n}$ by the corrected normal laws are considered in the model where the underlying distribution has a convolution structure. As a basic tool, the convergence part of Khinchine’s theorem in metric theory of Diophantine approximations is extended to the class of product characteristic functions.