MULTI-COLOUR BOUNDED REMAINDER SETS

Abstract

Let r(i, X1) be the number of points in the Sα-orbit of the length i with respect to a rotation Sα : T1 −→ T1 of the unit circle T1 = R/Z by an angle α hit the X1. Denote by δ(i, X1) = r(i, X1)−i|X1| the deviation of the function r(i, X1) from its average value i|X1|, where |X1| is the length of X1 . In 1921 E. Hecke had proved the theorem: if X1 has the length |X1| = hα + b, where h ∈ N, b ∈ Z, then the inequality |δ(i, X1)| � h для всех i = 0, 1, 2, . . . holds for all i = 0, 1, 2, . . . In 1981 г. I. Oren was able to generalize the Hecke theorem to the case of a finite union of intervals X1. He proved the estimation δ(i, X1) = O(1) as i → ∞. In the general case, if Xd belongs to the d-dimensional torus Td = Rd/Zd and there is δ(i, Xd) = O(1) as i → ∞, then Xd 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 Xd on the torus Td, k the complete toric decompositions Td = X0 d ⊔ Xd . . . ⊔ Xd with parameters c,λ 1 ⊔ s c, λ began to be considered. The main idea was to determine a lifting π−1 : R α maps d of the torus Td into the covering space Rd T ֒→ so the rotation S d ′ ′ ′ to a rearrangement Sv of the corresponding sets X0, X1, . . . , X in Rd. In the s ′ case s + 1 � d + 1, each set Xd = π(X ) is a bounded remainder set and the k k ′ ′ ′ union Td = X ⊔ X ⊔ . . . ⊔ X in Rd is a toric development for Td. These c,λ 0 1 s developments Td were built with the help of rearrangement parallelohedra, and the parallelohedra obtained as the Minkowskii sums of the unit cube Cd 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.