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Distribution of points of one-dimensional quasilattices with respect to a variable module. (English. Russian original) Zbl 1347.11017
Russ. Math. 56, No. 3, 14-19 (2012); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2012, No. 3, 17-23 (2012).
Summary: We consider one-dimensional quasiperiodic Fibonacci tilings. Namely, we study sets of vertices of these tilings that represent one-dimensional quasilattices defined on the base of a parameterization by rotations of a circle, and the distribution of points of quasilattices with respect to a variable module. We show that the distribution with respect to some modules is not uniform. We describe the distribution function and its integral representation, and estimate the remainder in the problem of the distribution of points of a quasilattice for corresponding modules.

MSC:
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B83 Special sequences and polynomials
05B45 Combinatorial aspects of tessellation and tiling problems
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[1] V. G. Zhuravlev, ”One-Dimensional Fibonacci Quasilattices and Their Applications to the Diophantine Equations and the Euclidean Algorithm,” Algebra i Analiz 19(3), 154–185 (2007).
[2] V.G. Zhuravlev, ”One-Dimensional Fibonacci Tilings,” Izv. Ross. Akad.Nauk, Ser.Matem. 71(2), 287–321 (2007). · Zbl 1168.11006
[3] P. Arnoux, V. Berthe, H. Ei, and S. Ito, ”Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions and Multidimensional Continued Fractions,” DiscreteModels: Combinatorics, Computation and Geometry (Paris, 2001), pp. 59–78 (electronic, only). · Zbl 1017.68147
[4] V. G. Zhuravlev, ”One-Dimensional Fibonacci Tilings and Derivatives of Two-Colour Rotations of a Circle,” Preprint Series 59, 1–43 (Max Plank-Institut für Mathematik, Bonn, 2004).
[5] N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes Math. (Springer, Berlin, 2002), Vol. 1794. · Zbl 1014.11015
[6] V. V. Krasil’shchikov, A. V. Shutov, and V. G. Zhuravlev, ”One-Dimensional Quasiperiodic Tilings Admitting Progressions Enclosure,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 3–9 (2009) [Russian Mathematics (Iz. VUZ) 53 (7), 1–6 (2009)]. · Zbl 1195.11084
[7] V. V. Krasil’shchikov and A. V. Shutov, ”Embedding of Lattices in Quasiperiodic Lattices,” in Studies in Algebra, Theory of Numbers, Functional Analysis, and Related Questions (Saratov, 2007), No. 4, 45–55.
[8] V. V. Krasil’shchikov and A. V. Shutov, ”Description and Exact Maximum and Minimum Values of the Remainder in the Problem of the Distribution of Fractional Parts,” Matem. Zametki 89(1), 43–52 (2011). · doi:10.4213/mzm5266
[9] V. V. Krasil’shchikov, ”The SpectrumofOne-DimensionalQuasilattices,” Sib. Matem. Zhurn. 51(1), 68–73 (2010).
[10] A. V. Shutov and A. V. Maleev, ”Quasiperiodic Plane Tilings Based on Stepped Surfaces,” Acta Crystallographica 64(3), 376–382 (2008). · Zbl 1370.52066 · doi:10.1107/S0108767308005059
[11] A. V. Shutov, A. V. Maleev, and V. G. Zhuravlev, ”Complex Quasiperiodic Self-Similar Tilings: Their Parameterization, Boundaries, Complexity, Growth and Symmetry,” Acta Crystallogrphica 66(3), 427–437 (2010). · doi:10.1107/S0108767310006616
[12] C. G. Pinner, ”On Sums of Fractional Parts n\(\alpha\) + \(\gamma\),” J. Number Theory 65(1), 48–73 (1997). · Zbl 0886.11045 · doi:10.1006/jnth.1997.2080
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