Rauzy fractals and their number-theoretic applications

In this paper, we construct and study Rauzy partitions of order $n$ for a certain class of Pisot numbers. These partitions are partitions of a torus into fractal sets. Moreover, the action of a certain shift of the torus on partitions introduced is reduced to rearranging the partition tiles. We obtain a number of applications of partitions introduced to the study of the corresponding shift of the torus. In particular, we prove that partition tiles are bounded-remainder sets with respect to the shift considered. In addition, we obtain a number of applications to the study of sets of positive integers that have a given ending of the greedy expansion by a linear recurrent sequence and to generalized Knuth–Matiyasevich multiplications.