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Izv. RAN. Ser. Mat., 2007, Volume 71, Issue 2, Pages 89–122 (Mi izv620)  

This article is cited in 24 scientific papers (total in 24 papers)

One-dimensional Fibonacci tilings

V. G. Zhuravlev

Vladimir State Pedagogical University

Abstract: We use the $B$-operator to construct a family of Fibonacci tilings $\operatorname{Til}(\varepsilon_m)$ of the unit interval $I_0=[0,1)$ consisting of $F_{m+1}$ short and $F_{m+2}$ long elementary intervals with the ratio of the lengths equal to the golden section $\tau=\frac{1+\sqrt{5}}2$. We prove that the tilings $\operatorname{Til}(\varepsilon_m)$ satisfy a recurrence relation similar to the relation $F_{m+2}=F_{m+1}+F_m$ for the Fibonacci numbers. The ends of the elementary intervals in the tilings $\operatorname{Til}(\varepsilon_m)$ form a sequence of points $O_0$ whose derivatives $d^mO_0 = O_0 \cap [1-\tau^{-m},1)$ are sequences $O_m$ similar to the sequence $O_0$. We compute the direct $R_m(i)$ and inverse $R_{-m}(i)$ renormalizations for the sequences $O_m$. We establish a connection between our tilings and the Sturm sequence, and give some applications of the tilings $\operatorname{Til}(\varepsilon_m)$ in the theory of numbers.

DOI: https://doi.org/10.4213/im620

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English version:
Izvestiya: Mathematics, 2007, 71:2, 307–340

Bibliographic databases:

UDC: 511
MSC: 68R15, 68Q45
Received: 19.11.2002
Revised: 28.02.2004

Citation: V. G. Zhuravlev, “One-dimensional Fibonacci tilings”, Izv. RAN. Ser. Mat., 71:2 (2007), 89–122; Izv. Math., 71:2 (2007), 307–340

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. G. Zhuravlev, “The attraction domain for the attractor of a two-color circle rotation”, J. Math. Sci. (N. Y.), 150:3 (2008), 2056–2083  mathnet  crossref  elib  elib
    2. V. G. Zhuravlev, “Arifmetika dvukhtsvetnykh povorotov okruzhnosti”, Chebyshevskii sb., 8:2 (2007), 56–72  mathnet  mathscinet  zmath
    3. V. G. Zhuravlev, “Even Fibonacci numbers: the binary additive problem, the distribution over progressions, and the spectrum”, St. Petersburg Math. J., 20:3 (2009), 339–360  mathnet  crossref  mathscinet  zmath  isi  elib
    4. V. G. Zhuravlev, “Two-colour rotations of the unit circle”, Izv. Math., 73:1 (2009), 79–120  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. V. V. Krasil'shchikov, A. V. Shutov, V. G. Zhuravlev, “One-dimensional quasiperiodic tilings admitting progressions enclosure”, Russian Math. (Iz. VUZ), 53:7 (2009), 1–6  mathnet  crossref  mathscinet  zmath  elib
    6. V. G. Zhuravlev, “One-dimensional Fibonacci tilings and induced two-colour rotations of the circle”, Izv. Math., 74:2 (2010), 281–323  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. V. G. Zhuravlev, “Parametrization of a two-dimensional quasiperiodic Rauzy tiling”, St. Petersburg Math. J., 22:4 (2011), 529–555  mathnet  crossref  mathscinet  zmath  isi
    8. V. G. Zhuravlev, “Hyperbolas over two-dimensional Fibonacci quasilattices”, J. Math. Sci., 182:4 (2012), 472–483  mathnet  crossref  mathscinet  elib
    9. V. G. Zhuravlev, “Geometrizatsiya teoremy Gekke”, Chebyshevskii sb., 11:1 (2010), 126–144  mathnet  mathscinet
    10. V. V. Krasil'shchikov, A. V. Shutov, “Distribution of points of one-dimensional quasilattices with respect to a variable module”, Russian Math. (Iz. VUZ), 56:3 (2012), 14–19  mathnet  crossref  mathscinet
    11. E. P. Davletyarova, A. A. Zhukova, A. V. Shutov, “Geometrization of Fibonacci numeration system and its applications to number theory”, St. Petersburg Math. J., 25:6 (2014), 893–907  mathnet  crossref  mathscinet  zmath  isi  elib
    12. A. V. Shutov, “Drobi Fareya i perestanovki, porozhdennye drobnymi dolyami $\{i\alpha\}$”, Chebyshevskii sb., 15:1 (2014), 195–203  mathnet
    13. V. G. Zhuravlev, “Imbedding of circular orbits and the distribution of fractional parts”, St. Petersburg Math. J., 26:6 (2015), 881–909  mathnet  crossref  mathscinet  isi  elib  elib
    14. A. A. Abrosimova, “$\mathrm{BR}$-mnozhestva”, Chebyshevskii sb., 16:2 (2015), 8–22  mathnet  elib
    15. D. V. Kuznetsova, A. V. Shutov, “Exchanged Toric Tilings, Rauzy Substitution, and Bounded Remainder Sets”, Math. Notes, 98:6 (2015), 932–948  mathnet  crossref  crossref  mathscinet  isi  elib
    16. V. G. Zhuravlev, “Two-dimension approximations by the method of dividing toric tilings”, J. Math. Sci. (N. Y.), 217:1 (2016), 54–64  mathnet  crossref  mathscinet
    17. V. G. Zhuravlev, “Dividing toric tilings and bounded remainder sets”, J. Math. Sci. (N. Y.), 217:1 (2016), 65–80  mathnet  crossref  mathscinet
    18. A. A. Osipova, “Vypuklye rombododekaedry i parametricheskie BR-mnozhestva”, Chebyshevskii sb., 17:1 (2016), 160–170  mathnet  elib
    19. V. G. Zhuravlev, “Symmetrization of bounded remainder sets”, St. Petersburg Math. J., 28:4 (2017), 491–506  mathnet  crossref  mathscinet  isi  elib
    20. V. G. Zhuravlev, “Bounded remainder sets”, J. Math. Sci. (N. Y.), 222:5 (2017), 585–640  mathnet  crossref  mathscinet
    21. E. P. Davletyarova, A. A. Zhukova, A. V. Shutov, “Geometrizatsiya obobschennykh sistem schisleniya Fibonachchi i ee prilozheniya k teorii chisel”, Chebyshevskii sb., 17:2 (2016), 88–112  mathnet  elib
    22. V. G. Zhuravlev, “Induced bounded remainder sets”, St. Petersburg Math. J., 28:5 (2017), 671–688  mathnet  crossref  mathscinet  isi  elib
    23. A. A. Zhukova, A. V. Shutov, “Geometrizatsiya sistem schisleniya”, Chebyshevskii sb., 18:4 (2017), 222–245  mathnet  crossref  elib
    24. V. G. Zhuravlev, “Yadernyi algoritm razlozheniya v mnogomernye tsepnye drobi”, Algebra i teoriya chisel. 1, Posvyaschaetsya pamyati Olega Mstislavovicha FOMENKO, Zap. nauchn. sem. POMI, 469, POMI, SPb., 2018, 32–63  mathnet
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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