Generalized Rauzy tilings and linear recurrence sequences

Rauzy introduced a fractal set associated with the two-dimensional toric shift by the vector $(\beta^{-1},\beta^{-2})$, where $\beta $ is the real root of the equation $\beta^3 = \beta^2 + \beta + 1 $ and showed that this fractal is divided into three fractals that are bounded remainder sets with respect to a given toric shift. The introduced set was named as Rauzy fractal. It obtains many applications in the combinatorics of words, geometry, theory of dynamical systems and number theory.
Later, an infinite sequence of tilings of $d-1$-dimensional Rauzy fractals associated with algebraic Pisot units of the degree $d$ into fractal sets of $d$ types were introduced. Each subsequent tiling is a subdivision of the previous one. These tilings are closely related to some irrational toric shifts and allowed to obtain new examples of bounded remainder sets for these shifts, and also to get some results on self-similarity of shift orbits.
In this paper, we continue the study of generalized Rauzy tilings related to Pisot numbers. A new approach to definition of Rauzy fractals and Rauzy tilings based on expansions of natural numbers on linear recurrence sequences is proposed. This allows to improve the results on the connection of Rauzy tilings and bounded remainder sets and to show that the corresponding estimate of the remainder term is independent on the tiling order.
The geometrization theorem for linear recurrence sequences is proved. It states that the natural number has a given endpoint of the greedy expansion on the linear recurrence sequence if and only if the corresponding point of the orbit of toric shift belongs to some set, which is the union of the tiles of the Rauzy tiling. Some number-theoretic applications of this result is obtained.
In conclusion, some open problems related to generalized Rauzy tilings are formulated.