Nonautonomous bounded remainder sets

Nonautonomous bounded remainder sets are sequences of sets admitting a uniform estimate of the remainder term in the distribution of the fractional parts of linear function problem. In the paper we give a complete description of nonautonomous bounded remainder sets in the case of periodic sequences. The result is also generalized to certain classes of quasiperiodic sequences of sets. The proofs are based on obtaining explicit formulas for the remainder term in the terms of sums of fractional parts. The method is effective, i. e., it allows us to obtain explicit estimates of the remainder term.