Substitutions and bounded remainder sets

The paper is devoted to the multidimensional problem of distribution of fractional parts of a linear function. A subset of a multidimensional torus is called a bounded remainder set if the remainder term of the multidimensional problem of the distribution of the fractional parts of a linear function on this set is bounded by an absolute constant. We are interested not only in the individual bounded remainder sets but also in toric tilings into such sets.
A new class of tilings of a $d$-dimensional torus into sets of $(d + 1)$ types is introduced. These tilings are defined in combinatorics and geometric terms and are called generalized exchanged tilings. It is proved that all generalized exchanged toric tilings consist of bounded remainder sets. Corresponding estimate of the remainder term is effective. We also find conditions that ensure that the estimate of the remainder term for the sequence of generalized exchanged toric tilings does not depend on the concrete tiling in the sequence.
Using the Arnoux-Ito theory of geometric substitutions we introduce a new class of generalized exchanged tilings of multidimensional tori into bounded remainder sets with an effective estimate of the remainder term. Earlier similar results were obtained in the two-dimensional case for one specific substitution — a geometric version of well-known Rauzy substitution. With the help of the passage to the limit, another class of generalized exchanged toric tilings into bounded remainder sets with fractal boundaries is constructed (so-called generalized Rauzy fractals).