Multi-colour bounded remainder sets

Let $r(i,X^1)$ be the number of points in the $S_{\alpha}$-orbit of the length $i$ with respect to a rotation $S_{\alpha}: \; \mathbb{T}^1 \longrightarrow \mathbb{T}^1$ of the unit circle $\mathbb{T}^1=\mathbb{R}/\mathbb{Z}$ by an angle $\alpha$ hit the $X^1$. Denote by $\delta(i,X^1)=r(i,X^1) - i|X^1|$ the deviation of the function $r(i,X^1)$ from its average value $i|X^1|$, where $|X^1|$ is the length of $X^1$.
In 1921 E. Hecke had proved the theorem: if $X^1$ has the length $|X^1|=h \alpha + b$, where $h\in \mathbb{N}$, $b\in \mathbb{Z}$, then the inequality $|\delta(i,X^1)|\le h$ для всех $i=0,1,2,\ldots$ holds for all $i=0,1,2,\ldots$
In 1981 г. I. Oren was able to generalize the Hecke theorem to the case of a finite union of intervals $X^1$. He proved the estimation $\delta(i,X^1) =O(1)$ as $i \rightarrow \infty$.
In the general case, if $X^d$ belongs to the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ and there is $\delta(i,X^d) =O(1)$ as $i \rightarrow \infty$, then $X^d$ is called a bounded remainder set.
Global approach to search of bounded remainder sets was proposed by V.G. Zhuravlev in 2011 when, instead of separate sets $X^d_k$ on the torus $\mathbb{T}^d$, the complete toric decompositions $\mathbb{T}^d_{c,\lambda}=X^d_0 \sqcup X^d_1\sqcup \ldots \sqcup X^d_{s}$ with parameters $c,\lambda$ began to be considered. The main idea was to determine a lifting $\pi^{-1}:\; \mathbb{T}^d \hookrightarrow \mathbb{R}^d$ of the torus $\mathbb{T}^d$ into the covering space $\mathbb{R}^d$ so the rotation $S_{\alpha}$ maps to a rearrangement $S_{v}$ of the corresponding sets $X'_0,X'_1, \ldots, X'_{s}$ in $\mathbb{R}^d$. In the case $s+1\le d+1$, each set $X^d_k=\pi(X'_k)$ is a bounded remainder set and the union $T^d_{c,\lambda}=X'_0 \sqcup X'_1 \sqcup \ldots \sqcup X'_s$ in $\mathbb{R}^d$ is a toric development for $\mathbb{T}^d$. These developments $T^d$ were built with the help of rearrangement parallelohedra, and the parallelohedra obtained as the Minkowskii sums of the unit cube $C^{d}$ and intervals. If $d=3,4$ we have the Voronoi parallelohedra and the Fedorov rhombic dodecahedron.
In the present paper, by using tilings of multidimensional tori, bounded remainder sets are constructed. The tilings consist of a finite combination of convex polyhedra. A multi-dimension version of Hecke theorem with respect to the uniform distribution of fractional parts on the unit circle is proved for these sets.
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