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St. Petersburg Mathematical Journal

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


Author: V. G. Zhuravlev
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 19 (2007), nomer 3.
Journal: St. Petersburg Math. J. 19 (2008), 431-454
MSC (2000): Primary 06A11
DOI: https://doi.org/10.1090/S1061-0022-08-01005-4
Published electronically: March 21, 2008
MathSciNet review: 2340709
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Abstract: The one-dimensional quasilattices $ \mathcal{L}$ lying in the square Fibonacci quasilattice $ \mathcal{F}^2=\mathcal{F} \times \mathcal{F}$ 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 $ X_i$ take values in $ \mathbb{N}=\lbrace 1,2,3,\ldots \rbrace $ and $ \circ$ is Knuth's circular multiplication.


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Additional Information

V. G. Zhuravlev
Affiliation: Vladimir State Pedagogical University, Prospekt Stroitelei 11, 600024, Vladimir, Russia
Email: vzhuravlev@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01005-4
Keywords: Fibonacci quasilattices, Diophantine equations, Knuth's circular multiplication
Received by editor(s): September 11, 2006
Published electronically: March 21, 2008
Additional Notes: Supported by RFBR (grant no. 05-01-00435)
Article copyright: © Copyright 2008 American Mathematical Society