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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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http://mi.mathnet.ru/eng/aa123 http://mi.mathnet.ru/eng/aa/v19/i3/p151
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:
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V. G. Zhuravlev, “The Pell equation over the $\circ$-Fibonacci ring”, J. Math. Sci. (N. Y.), 150:3 (2008), 2084–2095
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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
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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
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V. G. Zhuravlev, “Hyperbolas over two-dimensional Fibonacci quasilattices”, J. Math. Sci., 182:4 (2012), 472–483
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A. V. Shutov, “Arifmetika i geometriya odnomernykh kvazireshetok”, Chebyshevskii sb., 11:1 (2010), 255–262
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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
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A. V. Shutov, “Trigonometricheskie summy nad odnomernymi kvazireshetkami”, Chebyshevskii sb., 13:2 (2012), 136–148
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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
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A. A. Zhukova, A. V. Shutov, “Binarnaya additivnaya zadacha s chislami spetsialnogo vida”, Chebyshevskii sb., 16:3 (2015), 246–275
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V. G. Zhuravlev, “Symmetrization of bounded remainder sets”, St. Petersburg Math. J., 28:4 (2017), 491–506
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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
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A. A. Zhukova, A. V. Shutov, “Geometrizatsiya sistem schisleniya”, Chebyshevskii sb., 18:4 (2017), 222–245
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