Convex rhombic dodecahedron and parametric BR-sets

A. A. Osipova

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 160-170.

Résumé

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.

Détail

Keywords: bounded remainder sets, distribution of fractional parts, toric development.

Bibliographie

[1] Hecke E., “Eber Analytische Funktionen und die Verteilung von Zahlen mod. eins”, Math. Sem. Hamburg. Univ., 5 (1921), 54–76

[2] Erdös P., “Problems and results on diophantine approximation”, Comp. Math., 16 (1964), 52–65 | MR | Zbl

[3] Kesten H., “On a conjecture of Erdös and Szüsz related to uniform distribution mod 1”, Acta Arithmetica, 12 (1966), 193–212 | MR | Zbl

[4] Zhuravlev V. G., “One-dimensional Fibonacci tilings”, Izvestiya: Mathematics, 71:2 (2007), 89–122 | DOI | MR | Zbl

[5] Shutov A. V., “Optimum estimates in the problem of the distribution of fractional parts of the sequence $n\alpha$”, Vestnik SamGU. Yestestvennonauchnaya seriya, 5:3 (2007), 112–121

[6] Szüsz R., “Über die Verteilung der Vielfachen einer komplexen Zahl nach dem Modul des Einheitsquadrats”, Acta Math. Acad. Sci. Hungar., 1954, no. 5, 35–39 | DOI | MR | Zbl

[7] Zhuravlev V. G., “Rauzy tilings and bounded remainder sets on the torus”, Journal of Mathematical Sciences, 322 (2005), 83–106 | Zbl

[8] Zhuravlev V. G., “Exchanged toric developments and bounded remainder sets”, Journal of Mathematical Sciences, 392 (2011), 95–145

[9] Zhuravlev V. G., “Multidimensional Hecke theorem on the distribution of fractional parts”, St. Petersburg Mathematical Journal, 24:1 (2012), 1–33

[10] Abrosimova A. A., “Bounded remainder sets on a two-dimensional torus”, Chebyshevskiy sbornik, 12:4(40) (2011), 15–23 | MR

[11] Abrosimova A. A., “Average values for deviation distribution of points on the torus”, Belgorod State University Scientific bulletin. Mathematics Physics, 5(124):26 (2012), 5–11

[12] Abrosimova A. A., “Multiplication of toric developments and constructing of bounded remainder sets”, Uchenye zapiski Orlovskogo Gosudarstvennogo Universiteta. Seriya: Yestestvennyye, tekhnicheskiye i meditsinskiye nauki, 2012, no. 6/2, 30–37 | MR

[13] Abrosimova A. A., “Fractal bounded remainder sets”, Matematicheskoye modelirovaniye fraktal'nykh protsessov, rodstvennyye problemy analiza i informatiki" (II Int. Conf. Proc. of Young Scientists "Mathematical Modeling of Fractal Processes of Analysis and Informatics, Materialy II Mezhdunarodnoy konferentsii molodykh uchenykh, Nalchik, 2012, 18–21 | MR

[14] Abrosimova A. A., Blinov D. A., Polyakova T. V., “Optimization of boundaries of remainder for bounded remaider sets on two-dimensional torus”, Chebyshevskiy sbornik, 14:1(45) (2013), 9–17 | MR | Zbl

[15] Abrosimova A. A., “Boundaries of deviations for three-dimensional bounded remainder sets”, Belgorod State University Scientific bulletin. Mathematics Physics, 19(162):32 (2013), 5–21 | MR

[16] Abrosimova A. A., Blinov D. A., “Boundaries optimization of two-dimensional bounded remainder sets”, Belgorod State University Scientific bulletin. Mathematics Physics, 26(169):33 (2013), 5–13 | MR

[17] Abrosimova A. A., “BR-sets”, Chebyshevskiy sbornik, 16:2(54) (2015), 8–22 | MR | Zbl

[18] Weyl H., “Über die Gibbs'sche Erscheinung und verwandte Konvergenzph änomene”, Rendicontidel Circolo Mathematico di Palermo, 30 (1910), 377–407 | DOI | Zbl

[19] Knuth D., “Ecient balanced codes”, IEEE Trans. Inf. Theory, IT-32:1 (1986), 51–53 | DOI | MR | Zbl

[20] Altman E., Gaujual B., Hordijk A., “Balanced Sequences and Optimal Routing”, Journal of Association for Computing Machinery, 2000, no. 4, 752–775 | DOI | MR | Zbl

[21] Vuillon L., “Balanced words”, Bull. Belg. Math. Soc. Simon Stevin, 2003, no. 10, 787–805 | MR | Zbl

[22] Heinis A., “Languages under substitutions and balanced words”, J. de Theories des Nombres de Bordeaux, 2004, no. 16, 151–172 | DOI | MR | Zbl