Trigonometric Sums over One-Dimensional Quasilattices of Arbitrary Codimension

A. V. Shutov

Matematičeskie zametki, Tome 97 (2015) no. 5, pp. 781-793.

Résumé

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.

Détail

Keywords: trigonometric sum, codimension, bounded remainder set, tiling of the torus, Weyl's uniform distribution theorem, averaged lattice value, Koksma–Hlawka inequality
Mots-clés : quasilattice, orbit structure.

Bibliographie

[1] V. V. Krasilschikov, A. V. Shutov, “Odnomernye kvazikristally: approksimatsiya periodicheskimi strukturami i vlozhenie reshetok”, Noveishie problemy teorii polya, 5, Izd-vo Kazansk. un-ta, Kazan, 2006, 145–154

[2] V. V. Krasilschikov, A. V. Shutov, “Nekotorye voprosy vlozheniya reshetok v odnomernye kvaziperiodicheskie razbieniya”, Vestn. SamGU. Estestvennonauchn. ser., 2007, no. 7(57), 84–91

[3] V. V. Krasilschikov, A. V. Shutov, “Odnomernye kvaziperiodicheskie razbieniya, dopuskayuschie vlozhenie progressii”, Izv. vuzov. Matem., 2009, no. 7, 3–9 | MR | Zbl

[4] V. V. Krasilschikov, A. V. Shutov, “Raspredelenie tochek odnomernykh kvazireshetok po peremennomu modulyu”, Izv. vuzov. Matem., 2012, no. 3, 17–23 | MR

[5] V. V. Krasilschikov, “Spektr odnomernykh kvazireshetok”, Sib. matem. zhurn., 51:1 (2010), 68–73 | MR

[6] A. V. Shutov, “Arifmetika i geometriya odnomernykh kvazireshetok”, Chebyshevskii sb., 11:1 (2010), 255–262 | MR | Zbl

[7] A. V. Shutov, “Trigonometricheskie summy nad odnomernymi kvazireshetkami”, Chebyshevskii sb., 13:2 (2012), 136–148 | MR | Zbl

[8] I. M. Vinogradov, Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1971 | MR | Zbl

[9] C. Janot, Quasicrystals. A Primer, Oxf. Class. Texts Phys. Sci., Clarendon Press, Oxford, 1994 | MR | Zbl

[10] V. G. Zhuravlev, “Mnogomernaya teorema Gekke o raspredelenii drobnykh chastei”, Algebra i analiz, 24:1 (2012), 95–130 | MR | Zbl

[11] V. Baladi, D. Rockmore, N. Tongring, C. Tresser, “Renormalization on the $n$-dimensional torus”, Nonlinearity, 5:5 (1992), 1111–1136 | DOI | MR | Zbl

[12] G. Rauzy, “Nombres algébriques et substitutions”, Bull. Soc. Math. France, 110:2 (1982), 147–178 | MR | Zbl

[13] A. V. Shutov, “Dvumernaya problema Gekke–Kestena”, Chebyshevskii sb., 12:2 (2011), 151–162 | MR | Zbl

[14] A. V. Shutov, “Ob odnom semeistve dvumernykh mnozhestv ogranichennogo ostatka”, Chebyshevskii sb., 12:4 (2011), 264–271 | MR | Zbl

[15] N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math., 1794, Springer-Verlag, Berlin, 2001 | DOI | MR | Zbl

[16] V. G. Zhuravlev, “Mnogogranniki ogranichennogo ostatka”, Matematika i informatika, 1, K 75-letiyu so dnya rozhdeniya Anatoliya Alekseevicha Karatsuby, Sovr. probl. matem., 16, MIAN, M., 2012, 82–102 | DOI | Zbl

[17] A. V. Shutov, “Mnogomernye obobscheniya summ drobnykh dolei i ikh teoretiko-chislovye prilozheniya”, Chebyshevskii sb., 14:1 (2013), 104–118 | MR

[18] H. Weyl, “Über die Gibbs'sche Erscheinung und verwandte Konvergenzphänomene”, Palermo Rend., 30 (1910), 377–407 | DOI | Zbl

[19] C. Godrèche, C. Oguey, “Construction of average lattices for quasiperiodic structures by the section method”, J. Phys. France, 51 (1990), 21–37 | DOI | MR

[20] L. Keipers, G. Niderreiter, Ravnomernoe raspredelenie posledovatelnostei, Mir, M., 1985 | MR | Zbl

[21] V. G. Zhuravlev, “Chetno-fibonachchevy chisla: binarnaya additivnaya zadacha, raspredelenie po progressiyam i spektr”, Algebra i analiz, 20:3 (2008), 18–46 | MR | Zbl

[22] M. Drmota, R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Math., 1651, Springer-Verlag, Berlin, 1997 | DOI | MR | Zbl

[23] S. A. Stepanov, Arifmetika algebraicheskikh krivykh, Nauka, M., 1991 | MR | Zbl