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.
Bibliography: 9 titles.
