Multi-colour bounded remainder sets

V. G. Zuravlev

Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 93-116.

Résumé

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.

Détail

Keywords: multi-dimension Hecke theorem, bounded remainder sets, polyhedra.

Bibliographie

[1] Hecke E., “Über analytische Funktionen und die Verteilung von Zahlen mod. eins”, Math. Sem. Hamburg. Univ., 1 (1921), 54–76 | DOI | MR

[2] Oren I., “Admissible functions with multiple discontinuities”, Topology, Proc. spec. Semin., v. I, Univ. Nac. Autónoma México, Mexico City, 1981, 217–230 | MR

[3] Zhuravlev V. G., “A multidimensional Hecke theorem on the distribution of fractional parts”, St. Petersburg Math. J., 24:1 (2013), 71–97 | DOI | MR | Zbl

[4] Zhuravlev V. G., “Exchanged toric developments and bounded remainder sets”, J. Math. Sci. (N. Y.), 184:6 (2012), 716–745 | DOI | MR | Zbl

[5] Zhuravlev V. G., “Polyhedra bounded remainder”, Mathematics and informatics, 1, The 75th anniversary of Anatolia Alekseevicha Karatsuba, Sovrem. probl. Mat., 16, Steklov Mathematical Institute, M., 2012, 82–102 (Russian) | DOI

[6] Voronoi G. F., Collected works in three volumes, v. I, Izdatel'stvo Akademii Nauk Ukrainskoi SSR, Kiev, 1952, 399 pp.; v. II, 1952, 391 pp.; v. III, 1953, 306 pp. (Russian)

[7] Fedorov E. S., Elements of the study of figures, Izdat. Akad. Nauk SSSR, M., 1953, 410 pp. (Russian)

[8] Weyl H., “Über die Gleichverteilung von Zahlen $\mathrm{mod}$ Eins”, Math. Ann., 77 (1916), 313–352 | DOI | MR | Zbl

[9] Zhuravlev V. G., “Moduli of toric tilings into bounded remainder sets and balanced words”, St. Petersburg Math. J., 24:4 (2013), 601–629 | DOI | MR | Zbl

[10] Shutov A. V., “The Hecke–Kesten problem for some integrals”, Chebyshevskii Sb., 12:1(37) (2011), 172–177 (Russian) | MR | Zbl