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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



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
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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  • 1. A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. MR 0195803
  • 2. V. G. Zhuravlev, One-dimensional Fibonacci partitions, Proc. 17th Internat. Summer Workshop on Modern Problems of Theoretical and Mathematical Physics, Kazan', 2005, pp. 40-55. (Russian)
  • 3. V. G. Zhuravlev, Sums of squares over the Fibonacci o-ring, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 337 (2006), no. Anal. Teor. Chisel. i Teor. Funkts. 21, 165–190, 290 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 143 (2007), no. 3, 3108–3123. MR 2271962,
  • 4. George E. Cooke, A weakening of the Euclidean property for integral domains and applications to algebraic number theory. I, J. Reine Angew. Math. 282 (1976), 133–156. MR 0406973,
  • 5. N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. MR 1970385
  • 6. D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60. MR 0947168 (89f:11031)
  • 7. R. Lifshitz, The square Fibonacci tiling, J. Alloys Compounds 342 (2002), 186-190.
  • 8. Ju. V. Matijasevič, A connection between systems of word and length equations and Hilbert’s tenth problem, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 8 (1968), 132–144 (Russian). MR 0246772
  • 9. Ju. V. Matijasevič, Two reductions of Hilbert’s tenth problem, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 8 (1968), 145–158 (Russian). MR 0246773
  • 10. R. V. Moody, Model sets: A survey, Quasicrystals to More Complex Systems (F. Alex, F. Dénoyer, and J. P. Gazeau, eds.), EPD Science, Les Ulis, and Springer-Verlag, Berlin, 2000, pp. 145-166.

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Additional Information

V. G. Zhuravlev
Affiliation: Vladimir State Pedagogical University, Prospekt Stroitelei 11, 600024, Vladimir, Russia

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

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