zbMATH — the first resource for mathematics

The spectrum of one-dimensional quasilattices. (English. Russian original) Zbl 1209.11071
Sib. Math. J. 51, No. 1, 53-56 (2010); translation from Sib. Mat. Zh. 51, No. 1, 68-73 (2010).
The author proves theorems on the spectrum of the one-dimensional quasilattices resulting from irrational rotations of the circle.

11K31 Special sequences
11K36 Well-distributed sequences and other variations
Full Text: DOI
[1] Weyl H., ”Uniform distribution modulo 1,” in: H. Weyl, Selected Works [Russian translation], Nauka, Moscow, 1984, pp. 58–93.
[2] Korobov N. M., Trigonometric Series and Some of Its Applications [in Russian], Nauka, Moscow (1989). · Zbl 0665.10026
[3] Karatsuba A. A., ”Fractional parts of functions of a special form,” Izv.: Math., 59, No. 4, 721–740 (1995). · Zbl 0874.11050 · doi:10.1070/IM1995v059n04ABEH000031
[4] Karatsuba A. A., ”On fractional parts of rapidly growing functions,” Izv.: Math., 65, No. 4, 727–748 (2001). · Zbl 1028.11045 · doi:10.1070/IM2001v065n04ABEH000349
[5] Karatsuba A. A., Arkhipov G. I., and Chubarikov V. N., ”Distribution of fractional parts of polynomials in several variables,” Math. Notes, 25, No. 1, 3–9 (1979). · Zbl 0416.10038
[6] Kuipers L. and Niderreiter G., Uniform Distribution of Sequences, Wiley and Sons, New York (1974).
[7] Drmota M. and Tichy R. F., Sequences, Discrepancies and Applications, Springer-Verlag, Berlin (1997). · Zbl 0877.11043
[8] Arnoux P., Berthe V., Ei H., and Ito S., ”Tilings, quasicrystals, discrete planes, generalized substitutions and multidimensional continued fractions,” in: Discrete Models: Combinatorics, Computation and Geometry, Paris, 2001, pp. 59–78. · Zbl 1017.68147
[9] Zhuravlev V. G., One-Dimensional Fibonacci Tilings and Derivatives of Two-Colour Rotations of a Circle [Preprint / Max-Plank-Institut für Mathematik; No. 59], Leipzig (2004).
[10] Arnol’d V. I., ”Remarks on quasicrystallic symmetry,” in: F. Klein, Lectures on the Icosahedron and the Solution of the Fifth Degree [Russian translation], Nauka, Moscow, 1989, pp. 291–300.
[11] De Bruijn N. G., ”Sequences of zeros and ones generated by special production rules,” Kon. Nederl. Acad. Wetensch. Proc. Ser. A, 84, 38–52 (1982).
[12] Moody R. V., ”Model sets: a survey,” in: Quasicrystals to More Complex Systems, Les Houches, 1998 (F. Alex and J.-P. Gazeau, eds.), Centre de Physique des Houches, Springer-Verlag, Berlin, 2000, 13, pp. 145–166.
[13] Fogg N. P., Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin and Heidelberg (2002). · Zbl 1014.11015
[14] Zhuravlev V. G., ”Even Fibonacci numbers: the binary additive problem, the distribution over progressions, and the spectrum,” St. Petersburg Math. J., 20, No. 3, 339–360 (2009). · Zbl 1206.11020 · doi:10.1090/S1061-0022-09-01051-6
[15] Krasil’shchikov V. V. and Shutov A. V., ”On the distribution of sequences by a variable modulus,” in: Proceedings of the XXVIII Conference of Young Scientists of the Department of Mechanics and Mathematics, Moscow Univ., Moscow, 2006, pp. 90–93.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.