Exchanged Toric Tilings, Rauzy Substitution, and Bounded Remainder Sets

This paper is devoted to the two-dimensional problem of the distribution of the fractional parts of a linear function. A new class of tilings of the two-dimensional torus into bounded remainder sets with an effective estimate of the remainder is introduced. It is shown that examples of the tilings under consideration can be obtained by using the geometric version of the Rauzy substitution.