Geodesic
One-dimensional quasiperiodic tilings admitting progressions enclosure
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2009), pp. 3-9.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider one-dimensional quasiperiodic tilings based on the use of irrational rotations of a circle. We completely describe a wide class of progressions included in the mentioned tilings.
Keywords: one-dimensional quasiperiodic tilings, lattice enclosure. 
@article{IVM_2009_7_a0,
     author = {V. V. Krasil'shchikov and A. V. Shutov and V. G. Zhuravlev},
     title = {One-dimensional quasiperiodic tilings admitting progressions enclosure},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--9},
     publisher = {mathdoc},
     number = {7},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_7_a0/}
}
TY  - JOUR
AU  - V. V. Krasil'shchikov
AU  - A. V. Shutov
AU  - V. G. Zhuravlev
TI  - One-dimensional quasiperiodic tilings admitting progressions enclosure
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2009
SP  - 3
EP  - 9
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2009_7_a0/
LA  - ru
ID  - IVM_2009_7_a0
ER  - 
%0 Journal Article
%A V. V. Krasil'shchikov
%A A. V. Shutov
%A V. G. Zhuravlev
%T One-dimensional quasiperiodic tilings admitting progressions enclosure
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2009
%P 3-9
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2009_7_a0/
%G ru
%F IVM_2009_7_a0
V. V. Krasil'shchikov; A. V. Shutov; V. G. Zhuravlev. One-dimensional quasiperiodic tilings admitting progressions enclosure. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2009), pp. 3-9. http://geodesic.mathdoc.fr/item/IVM_2009_7_a0/

[1] Moody R. V., “Model sets: a survey”, From quasicrystals to more complex systems, eds. F. Alex, F. Dénoyer, J. P. Gazeau, EPD Science, Les Ulis; Springer-Verlag, Berlin, 2000, 145–166

[2] Fogg N. Pytheas, Substitutions in dynamics, arithmetics and combinatorics, Springer, 2002, 402 pp. | MR | Zbl

[3] Baake M., Schlottmann M., “Geometric aspects of tilings and equivalence concepts”, Proc. Fifth Conf. quasiqrystals, Singapore, 1995, 15–21

[4] de Bruijn N. G., “Algebraic theory of Penrose's non-periodic tilings of the plane”, Math. Proc. A, 84 (1981), 39–52

[5] Duneau M., Katz A., “Quasiperiodic patterns”, Phys. Rev. Lett., 54 (1985), 2688–2691 | DOI | MR

[6] Zhuravlev V. G., “Odnomernye razbieniya Fibonachchi. Noveishie problemy teorii polya”, Tr. XVII Mezhdunarodn. letnei shkoly-seminara “ Volga-17' 05” po sovremen. probl. teor. i matem. fiz., T. 5, Izd-vo KGU, Kazan, 2006, 40–55

[7] Zhuravlev V. G., “Razbieniya Rozi i mnozhestva ogranichennogo ostatka”, Zap. nauchn. semin. POMI, 322, 2005, 83–106 | MR | Zbl

[8] Zhuravlev V. G., Maleev A. V., “Posloinyi rost kvaziperiodicheskogo razbieniya Rozi”, Kristallografiya, 52:1 (2007), 204–210

[9] Zhuravlev V. G., “Odnomernye razbieniya Fibonachchi”, Izv. RAN. Cer. matem., 71:2 (2007), 89–122 | MR | Zbl

[10] Zhuravlev V. G., “Odnomernye kvazireshetki Fibonachchi i ikh prilozheniya k diofantovym uravneniyam i algoritmu Evklida”, Algebra i analiz, 19:3 (2007), 151–182 | MR

[11] Zhuravlev V. G., “One-dimensional Fibonacci tilings and derivatives of two-colour rotations of a circle”, Max-Plank-Institut für Mathematik Preprint Series, 59 (2004), 1–43 | MR

[12] Krasilschikov V. V., Shutov A. V., “Odnomernye kvazikristally: approksimatsiya periodicheskimi strukturami i vlozhenie reshetok. Noveishie problemy teorii polya”, Tr. XVII Mezhdunarodn. letnei shkoly-seminara “Volga-17' 05” po sovremen. probl. teor. i matem. fiz., T. 5, Izd-vo KGU, Kazan, 2006, 145–154

[13] Krasilschikov V. V., Shutov A. V., “O nekotorykh svoistvakh odnomernykh kvazikristallov”, Tez. dokl. XVIII Mezhdunarodn. letnei shkoly-seminara “Volga-18' 06” po sovremen. probl. teor. i matem. fiz., Kazan, 2006, 45–46

[14] Hecke E., “Über Analytische Funktionen und die Verteilung der Zahlen modulo Eins”, Math. Sem. Hamburg Univ., 1:1 (1922), 54–76 | DOI | MR

[15] Kesten H., “On a conjecture of Erdős and Szüsz related to uniform distribution $\mathrm{mod}1$”, Acta Arith., 12 (1966), 193–212 | MR | Zbl

[16] Ostrowski A., “Notiz zur Theorie der Diophantischen Approximationen und zur Theorie der linearen Diophantischen Approximationen”, Math. Miszelleren XVI, Jahresber. Deutschen Math. Ver., 39 (1939), 34–46

[17] Shutov A. V., “Sistemy schisleniya i mnozhestva ogranichennogo ostatka”, Tr. konf. “Analiticheskie i kombinatornye metody v teorii chisel”, M., 2006, 106–115