$\mathrm{BR}$-sets

Vladimir school of number theory was studied quasiperiodic tilings for a long time. The uniform distribution problem of fractional parts on the torus come from here. It is important to find exact estimates of the remainder for this distribution.
The paper is devoted to the important problem of number theory: bounded remainder sets. Relevance of the problem caused by the transition from the classical numerical and functional arithmetic structures to nonlinear arithmetic structures. Dynamical systems on bounded remainder sets generate balanced words,similar to words Sturmian and Rauzy words. Balanced words are important, for dynamical systems, coding theory,theory of communications and optimization problems, theory of languages and linguistics, recognition theory, statistical physics, etc.
The purpose of our research is construction of multidimensional bounded remainder sets and finding exact estimates of the remainder for this sets. The solution to this problem we start from two-dimensional case. We construct three classes of three-parameter two-dimensional bounded remainder sets. For their construction, we use hexagonal toric development. Now we know bounded remainder intervals, obtained by Hecke, and two-dimensional bounded remainder sets. There is the question: can we construct a new multi-dimensional sets using known sets? We construct four classes of four-parameter three-dimensional bounded remainder sets. We used for this the multiplication of toric developments. By multiplication of Hecke's intervals and two-dimensional hexagonal   developments   we   obtain   three-dimensional   hexagonal Fedorov's prisms-developments. For all described sets we give exact estimates of the remainder and prove generalization of Hecke's theorem to the multidimensional case. Also we obtain average values of the remainders, and fined sets with minimal value of the remainder.
This paper is an expository of the author's main results on bounded remainder sets.
Bibliography: 26 titles.