$n$-crowns in toric tilings into bounded remander sets

A. A. Zhukova ; A. V. Shutov

Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 246-260.

Résumé

The Arnoux-Ito theory of geometric substitutions allows to construct sequences of generalized exchanged tilings of the $d$-dimensional torus. These tilings consist of parallelepipeds of $d+1$ type, and the action of a certain toric shift on the tiling reduces to exchanging of the $d+1$ central parallelepipeds. Moreover, the set of vertices of all parallelepipeds of the tiling is a fragment of the orbit of zero point under considered toric shift. The considered tilings are actively used in various problems of number theory, combinatorics, and the theory of dynamical systems.In this paper, we study the local structure of toric tilings obtained using geometric substitutions. The $n$-corona of the parallelepiped is a set of all parallelepipeds located at a distance of not greater than $n$ from a given parallelepiped in the natural metric of the tiling. The problem is to describe all possible types of $n$-coronas.With each parallelepiped in the tiling we can naturally assigned a number — its number in the orbit of the corresponding central parallelepiped with respect to the toric shift. It is proved that the set of all parallelepipeds numbers splits into a finite number of half-intervals defining possible types of $n$-coronas. Moreover, it is proved that the boundaries of the corresponding half-open intervals are determined by the numbers of the parallelepipeds in the $n$-corona of the set of $d+1$ central parallelepiped.It is shown that this result can be considered as some multi-dimensional generalization of the famous three lengths theorem. Earlier, a similar description was obtained for 1-coronas of the toric tilings obtained using one specific geometric substitution: the Rauzy substitution. In addition, similar results were previously obtained for some quasiperiodic plane tilings.In conclusion, some directions for further research are formulated.

Détail

Keywords: geometric substitutions, Arnoux-Ito theory, generalized exchanged toric tiling, local structure, $n$-corona.

Bibliographie

[1] Arnoux P., Ito S., “Pisot substitutions and Rauzy fractals”, Bull. Belg. Math. Soc. Simon Stevin, 8:2 (2001), 181–207 | MR | Zbl

[2] Floreik K., “Une remarque sur la repartition des nombres $m\xi\bmod 1$”, Coll. Math. Wroclaw, 2 (1951), 323–324

[3] Lagarias J. C., Pleasants P. A. B., “Repetitive Delone sets and quasicrystals”, Ergod. Th. Dyn. Sys., 23 (2003), 831–867 | DOI | MR | Zbl

[4] Pytheas Fogg N., Substitutions in dynamics, arithmetics and combinatorics, Springer, 2001 | DOI | MR

[5] Rauzy G., “Nombres algébriques et substitutions”, Bull. Soc. Math. France, 110 (1982), 147–178 | DOI | MR | Zbl

[6] Ravenstein T. V., “The three gap theorem (Steinhaus conjecture)”, J. Austral. Math. Soc. Ser. A, 45 (1988), 360–370 | MR | Zbl

[7] Schattschneider D., Dolbilin N., “One corona is enough for the Euclidean plane”, Quasicrystals and discrete geometry (Toronto, ON, 1995), Fields Inst. Monogr., 10, Amer. Math. Soc., Providence. RI, 1998, 207–246 | MR | Zbl

[8] Shutov A. V., Maleev A. V., “Quasiperiodic planetilings based on stepped surfaces”, Acta Crystallographica. Section A: Foundations of Crystallography, 64:3 (2008), 376–382 | DOI | MR | Zbl

[9] Shutov A. V., Maleev A. V., Zhuravlev V. G., “Complex quasiperiodic self-similar tilings: Their parameterization, boundaries, complexity, growth and symmetry”, Acta Crystallographica. Section A: Foundations of Crystallography, 66:3 (2010), 427–437 | DOI | MR

[10] Siegel A., Thuswaldner J. M., “Topological properties of Rauzy fractals”, Mem. Soc. Math. Fr. N.S., 118 (2009), 1–144 | DOI | MR

[11] Slater N., “Gaps and steps for the sequence $n\theta$ mod 1”, Proc. Camb. Phil. Soc., 63 (1967), 1115–1123 | DOI | MR | Zbl

[12] Sós V. T. S., “On the distribution mod 1 of the sequence $n\alpha$”, Ann. Univ. Sci. Budapest Eötvös Sect. Math., 1 (1958), 127–134 | Zbl

[13] Świerczkowski S., “On successive settings of an arc on the circumference of a circle”, Fund. Math., 46 (1958), 187–189 | MR

[14] Zhuravlev V. G., “On additivity property of the complexity function related to rauzy tiling”, Analytic and Probabilistic Methods in Number Theory, 2007, 240–254 | MR | Zbl

[15] Zhukova A. A., Shutov A. V., “Rauzy substitution and local structure of torus tilings”, Chebyshevskii Sbornik, 2019:4 | Zbl

[16] Zhuravlev V. G., “Two-dimensional approximations by the method of dividing toric tilings”, Journal of Mathematical Sciences, 217:1 (2016), 54–64 | DOI | MR | Zbl

[17] Zhuravlev V. G., “Dividing toric tilings and bounded remainder sets”, Journal of Mathematical Sciences, 217:1 (2015), 65–80 | DOI

[18] Zhuravlev V. G., “Differentiation of induced toric tilings and multi-dimensional approximations of algebraic numbers”, Journal of Mathematical Sciences, 222:5 (2016), 544–584 | DOI | MR

[19] Zhuravlev V. G., “Induced bounded remainder sets”, St. Petersburg Math. J., 28 (2017), 671–688 | DOI | MR | Zbl

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

[21] Zhuravlev V. G., “One-dimensional Fibonacci tilings”, Izvestiya: Mathematics, 71:2 (2007), 281–323 | DOI | MR | Zbl

[22] Zhuravlev V. G., “Rauzy tilings and bounded remainder sets on the torus”, Journal of Mathematical Sciences, 137:2 (2006), 4658–4672 | DOI | MR | Zbl

[23] Zhuravlev V. G., Maleev A. V., “Layer-by-layer growth of quasi-periodic Rauzy tiling”, Crystallography Reports, 52:2 (2007), 180–186 | DOI

[24] Zhuravlev V. G., Maleev A. V., “Complexity function and forcing in the 2D quasi-periodic Rauzy tiling”, Crystallography Reports, 52:4 (2007), 582–588 | DOI

[25] Krasilshchikov V. V., Shutov A. V., One-dimensional quasiperiodic tilings and their applications, VF RUK, Vladimir, 2011

[26] Kuznetsova D. V., Shutov A. V., “Exchanged toric tilings, Rauzy substitution, and bounded remainder sets”, Mathematical Notes, 98:5–6 (2015), 932–948 | DOI | Zbl

[27] Manuylov N. N., “Self-similarity of some sequences of points on a circle”, Journal of Mathematical Sciences, 129:3 (2005), 3860–3867 | DOI | MR

[28] Manuylov N. N., “Direct renormalizations on the one-dimensional torus”, Journal of Mathematical Sciences, 133:6 (2006), 1686–1692 | DOI | MR | Zbl

[29] Shutov A. V., “Shifting on the torus and the multidimensional Hecke-Kesten problem”, Scientific notes of Oryol State University, 6:2 (2012), 249–253

[30] Shutov A. V., “Substitutions and bounded remainder sets”, Chebyshevskii Sbornik, 19:2 (2018), 501–522 | DOI | MR | Zbl

[31] Shutov A. V., “Derivatives of circle rotations and similarity of orbits”, Journal of Mathematical Sciences, 133:6 (2006), 1765–1771 | DOI | MR | Zbl

[32] Shutov A. V., “Number systems and bounded remainder sets”, Chebyshevskii Sbornik, 7:3 (2006), 110–128 | MR | Zbl

[33] Shutov A. V., Maleev A. V., “Strong parameterization and coordination encirclements of graph of Penrose tiling vertices”, Crystallography Reports, 62:4 (2017), 535–542 | DOI