Abstract
A new class of one-dimensional quasilattices parametrized by the translations of the torus is introduced. For this class, parameter-dependent trigonometric sums over points of quasilattice are considered. Nontrivial estimates of the trigonometric sums under consideration are obtained. For a number of trigonometric sums of special form, asymptotic formulas are derived. It is proved that the distribution of points of quasilattices is uniform modulo h for almost all h. Earlier similar results were obtained in the particular case of quasilattices parametrized by the rotations of the circle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. V. Krasil’shchikov and A. V. Shutov, “One-dimensional quasicrystals: Approximation by periodic structures and embedding of lattices,” in Newest Problems of Field Theory (Izd. Kazan Univ., Kazan, 2006), Vol. 5, pp. 145–154 [in Russian].
V. V. Krasil’shchikov and A. V. Shutov, “Some questions of the embedding of lattices in one-dimensional quasiperiodic tilings,” Vestnik SamGU, Estestvennonauchn. Ser., No. 7 (57), 84–91 (2007).
V. V. Krasil’shchikov and A. V. Shutov, “One-dimensional quasiperiodic tilings admitting progressions enclosure,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 3–9 (2009) [Russian Math. (Iz. VUZ) 53 (7), 1–6 (2009)].
V. V. Krasil’shchikov and A. V. Shutov, “Distribution of points of one-dimensional quasilattices with respect to a variable module,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 17–23 (2012) [Russian Math. (Iz. VUZ) 56 (3), 14–19 (2012)].
V. V. Krasil’shchikov, “The spectrum of one-dimensional quasilattices,” Sibirsk. Mat. Zh. 51(1), 68–73 (2010) [Siberian Math. J. 51 (1), 53–56 (2010)].
A. V. Shutov, “The arithmetic and geometry of one-dimensional quasilattices,” Chebyshevskii Sb. 11(1), 255–262 (2010).
A. V. Shutov, “Trigonometric sums over one-dimensional quasilattices,” Chebyshevskii Sb. 13(2), 136–148 (2012).
I.M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers (Nauka, Moscow, 1971; Dover Publications, 2004).
C. Janot, Quasicrystals: A Primer, in Oxford Class. Texts Phys. Sci. (Clarendon Press, Oxford, 1994).
V. G. Zhuravlev, “Multidimensional Hecke theorem on the distribution of fractional parts,” Algebra Anal. 24(1), 95–130 (2012) [St. Petersburg Math. J. 24 (1), 71–97 (2013)].
V. Baladi, D. Rockmore, N. Tongring, and C. Tresser, “Renormalization on the n-dimensional torus,” Nonlinearity 5(5), 1111–1136 (1992).
G. Rauzy, “Nombres algébriques et substitutions,” Bull. Soc. Math. France 110(2), 147–178 (1982).
A. V. Shutov, “The two-dimensional Hecke-Kesten problem,” Chebyshevskii Sb. 12(2), 151–162 (2011).
A. V. Shutov, “On a family of the two-dimensional bounded remainder sets,” Chebyshevskii Sb. 12(4), 264–271 (2011).
N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics, and Combinatorics, in Lecture Notes in Math. (Springer-Verlag, Berlin, 2001), Vol. 1794.
V.G. Zhuravlev, “Bounded remainder polyhedra,” in Trudy Mat. Inst. Steklov, Sovrem. Probl. Mat. (MIAN, Moscow, 2012), Vol. 16, pp. 82–102 [Proc. Steklov Inst. Math. 280, Suppl. 2, S71–S90 (2013)].
A. V. Shutov, “Multidimensional generalizations of sums of fractional parts and their number-theoretic applications,” Chebyshevskii Sb. 14(1), 104–118 (2013).
H. Weyl, “Über die Gibbs’sche Erscheinung und verwandte Konvergenzph änomene,” Palermo Rend. 30, 377–407 (1910).
C. Godrèche and C. Oguey, “Construction of average lattices for quasiperiodic structures by the section method,” J. Phys. France 51, 21–37 (1990).
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (Interscience, New York-London-Sydney, 1974; Nauka, Moscow, 1985).
V. G. Zhuravlev, “Fibonacci-even numbers: Binary additive problem, distribution over progressions, and spectrum,” Algebra Anal. 20(3), 18–46 (2008) [St. Petersburg Math. J. 20 (3), 339–360 (2009)].
M. Drmota and R. F. Tichy, Sequences, Discrepancies, and Applications, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1997), Vol. 1651.
S. A. Stepanov, Arithmetic of Algebraic Curves (Nauka, Moscow, 1991; Springer-Verlag, 1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. V. Shutov, 2015, published in Matematicheskie Zametki, 2015, Vol. 97, No. 5, pp. 781–793.
Rights and permissions
About this article
Cite this article
Shutov, A.V. Trigonometric sums over one-dimensional quasilattices of arbitrary codimension. Math Notes 97, 791–802 (2015). https://doi.org/10.1134/S0001434615050144
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434615050144