Local discrepancies in the problem linear function fractional parts distribution

In this paper we consider a problem of distribution of the fractional parts of the sequence obtained as multiplies of some irrational number with bounded partial quotients of its continued fraction expansion. Local discrepancies are the remainder terms of asymptotic formulas for the number of points of the sequence lying in given intervals. Earlier only intervals with bounded and logariphmic local discrepancies were known. We prove that there exists an infinite set of intervals with arbitrary small growth of local discrepancies. The proof is based on the connection of considered prolem with some problems from diophantine approximations.