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Izvestiya: Mathematics, 2007, Volume 71, Issue 2, Pages 307–340
DOI: https://doi.org/10.1070/IM2007v071n02ABEH002358
(Mi im620)
 

This article is cited in 42 scientific papers (total in 42 papers)

One-dimensional Fibonacci tilings

V. G. Zhuravlev

Vladimir State Pedagogical University
References:
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.
Received: 19.11.2002
Revised: 28.02.2004
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2007, Volume 71, Issue 2, Pages 89–122
DOI: https://doi.org/10.4213/im620
Bibliographic databases:
UDC: 511
MSC: 68R15, 68Q45
Language: English
Original paper language: Russian
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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\paper One-dimensional Fibonacci tilings
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\pages 89--122
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\jour Izv. Math.
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\pages 307--340
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Linking options:
  • https://www.mathnet.ru/eng/im620
  • https://doi.org/10.1070/IM2007v071n02ABEH002358
  • https://www.mathnet.ru/eng/im/v71/i2/p89
  • This publication is cited in the following 42 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:995
    Russian version PDF:339
    English version PDF:17
    References:75
    First page:4
     
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