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Algebra i Analiz, 2007, Volume 19, Issue 3, Pages 151–182 (Mi aa123)  

This article is cited in 12 scientific papers (total in 12 papers)

Research Papers

One-dimensional Fibonacci quasilattices and their application to the Euclidean algorithm and Diophantine equations

V. G. Zhuravlev

Vladimir State Pedagogical University

Abstract: The one-dimensional $\mathcal{L}$ quasilattices $\mathcal{F}^2=\mathcal{F}\times\mathcal{F}$ lying in the square Fibonacci quasilattice are classified; here $\mathcal{F}$ is the one-dimensional Fibonacci quasilattice. It is proved that there exists a countable set of similarity classes of quasilattices $\mathcal{L}$ in $\mathcal{F}^2$ (fine classification), and also four classes of local equivalence (rough classification).
Asymptotic distributions of points in quasilattices $\mathcal{L}$ are found and then applied to Diophantine equations involving the function $[\alpha]$ (the integral part of $\alpha$) and to equations of the form $A_1\circ X_1-A_2\circ X_2=C$ where the coefficients $C$ and $A_i$ and the variables take values in $\mathbb {N}=\{1,2,3,…\}$ and $\circ$ is Knuth's circular multiplication.

Keywords: Fibonacci quasilattices, Diophantine equations, Knuth's circular multiplication.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:3, 431–454

Bibliographic databases:

MSC: 06A11
Received: 11.09.2006

Citation: V. G. Zhuravlev, “One-dimensional Fibonacci quasilattices and their application to the Euclidean algorithm and Diophantine equations”, Algebra i Analiz, 19:3 (2007), 151–182; St. Petersburg Math. J., 19:3 (2008), 431–454

Citation in format AMSBIB
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\paper One-dimensional Fibonacci quasilattices and their application to the Euclidean algorithm and Diophantine equations
\jour Algebra i Analiz
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\vol 19
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\pages 151--182
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\zmath{https://zbmath.org/?q=an:1219.11033}
\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 3
\pages 431--454
\crossref{https://doi.org/10.1090/S1061-0022-08-01005-4}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. G. Zhuravlev, “The Pell equation over the $\circ$-Fibonacci ring”, J. Math. Sci. (N. Y.), 150:3 (2008), 2084–2095  mathnet  crossref
    2. 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
    3. 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
    4. V. G. Zhuravlev, “Hyperbolas over two-dimensional Fibonacci quasilattices”, J. Math. Sci., 182:4 (2012), 472–483  mathnet  crossref  mathscinet  elib
    5. A. V. Shutov, “Arifmetika i geometriya odnomernykh kvazireshetok”, Chebyshevskii sb., 11:1 (2010), 255–262  mathnet  mathscinet
    6. 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
    7. A. V. Shutov, “Trigonometricheskie summy nad odnomernymi kvazireshetkami”, Chebyshevskii sb., 13:2 (2012), 136–148  mathnet
    8. 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
    9. A. A. Zhukova, A. V. Shutov, “Binarnaya additivnaya zadacha s chislami spetsialnogo vida”, Chebyshevskii sb., 16:3 (2015), 246–275  mathnet  elib
    10. V. G. Zhuravlev, “Symmetrization of bounded remainder sets”, St. Petersburg Math. J., 28:4 (2017), 491–506  mathnet  crossref  mathscinet  isi  elib
    11. 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
    12. A. A. Zhukova, A. V. Shutov, “Geometrizatsiya sistem schisleniya”, Chebyshevskii sb., 18:4 (2017), 222–245  mathnet  crossref  elib
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