Convex rhombic dodecahedron and parametric BR-sets

The paper is devoted to the important problem of number theory: bounded remainder sets.
We consider the point orbits on low-dimensional tori. Any starting point generates the orbit under an irrational shift of the torus. The orbit is everywhere dense and uniformly distributed on the torus if the translation vector is irrational. Denote by $r(i)$ a function that gives the number of the orbit points which get some domain $T$. Then we have the formula $ r(i) = i \: \mathrm{ vol} (T) + \delta(i)$, where $\delta(i)=o(i)$ is the remainder. If the boundaries of the remainder are limited by a constant, then $T$ is a bounded remainder set (BR-set).
The article introduces a new BR-sets construction method, it is based on tilings parametric polyhedra. Сonsidered polyhedra are the torus development. Torus development should be to tile into figures, that can be exchanged, and we again obtain our torus development. This figures exchange equivalent shift of the torus.
Author have constructed tillings with this property and two-dimensional BR-sets. The considered method gives exact estimates and the average value of the remainder. Also we obtain the optimal BR-sets which have minimal values of the remainder. These BR-sets generate the strong balanced words (a multi-dimensional analogue of the Sturmian words).
The above method is applied to the case of three-dimensional torus in this paper. Also we obtain exact estimates and the average value of the remainder for constructed sets.
Bibliography: 22 titles.