Central Limit Theorem and Diophantine Approximations

Let F-n denote the distribution function of the normalized sum Z(n) = (X-1+ ... +X-n)/(sigma root n) of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of F-n to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of F-n by the Edgeworth corrections (modulo logarithmically growing factors in n), are given in terms of the characteristic function of X-1. Particular cases of the problem are discussed in connection with Diophantine approximations.