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Moments for the multidimensional Mönkemeyer algorithm


Author: M. Yu. Vodolagin
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 25 (2013), nomer 4.
Journal: St. Petersburg Math. J. 25 (2014), 533-545
MSC (2010): Primary 11J70
DOI: https://doi.org/10.1090/S1061-0022-2014-01305-2
Published electronically: June 5, 2014
MathSciNet review: 3184615
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Abstract | References | Similar Articles | Additional Information

Abstract: Moments asymptotics is studied for the partition corresponding to the multidimensional Mönkemeyer algorithm. A multidimensional generalization of a two-dimensional result by Moshchevitin and Vielhaber is proved.


References [Enhancements On Off] (What's this?)

  • 1. A. J. Brentjes, Multidimensional continued fraction algorithms, Mathematical Centre Tracts, vol. 145, Mathematish Centrum, Amsterdam, 1981. MR 638474 (83b:10038)
  • 2. A. Brocot, Calcul des rouages par approximation, nouvelle méthode, Revue Chronométrique 6 (1860), 186-194.
  • 3. D. J. Grabiner, Farey nets and multidimensional continued fractions, Monatsh. Math. 114 (1992), no. 1, 35-61. MR 1184802 (94b:11009)
  • 4. A. Hurwitz, Über die angenäherte Darstellung der Zahlen durch rationale Brüche, Math. Ann. 44 (1894), 417-436. MR 1510845
  • 5. J. C. Lagarias, Number theory and dynamical systems. The unreasonable effectivness of number theory (Orono, ME, 1991), Proc. Symp. Appl. Math., vol. 46, Amer. Math. Soc., Providence, RI, 1992. MR 1195841 (93m:11143)
  • 6. R. Mönkemeyer, Über Fareynetze in Dimensionen, Math. Nachr, 11 (1954), 321-344. MR 0064084 (16:223b)
  • 7. N. Moshchevitin and A. Zhigljavsky, Entropies of the partitions of the unit interval generated by the Farey tree, Acta Arith. 115 (2004), no. 1, 47-58. MR 2102805 (2005g:11145)
  • 8. N. Moshchevitin and M. Vielhaber, Moments for generalized Farey-Brocot partitions, Funct. Approx. Comment. Math. 38 (2008), no. 2, 131-157. MR 2492853 (2010c:11079)
  • 9. T. Prellberg and J. Slawny, Maps of intervals with indifferent fixed points: Thermodynamic formalism and phase transitions, J. Statist. Phys. 66 (1992), no. 1-2, 503-514. MR 1149493 (93g:58085)
  • 10. F. Schweiger, Multidimensional continued fractions, Oxford Univ. Press, Oxford, 2000. MR 2121855 (2005i:11090)
  • 11. M. Stern, Über eine Zahlentheoretische Funktion, Crelles J. Reine Angew. Math. 55 (1858), 193-220.
  • 12. M. Vielkhaber and N. Moshchevitin, Asymptotics for two-dimensional Farey-Brocot nets, Dokl. Akad. Nauk 416 (2007), no. 1, 11-14; English transl., Dokl. Math. 76 (2007), no. 2, 645-648. MR 2458591
  • 13. A. A. Dushistova, On the partitioning of the interval $ [0, 1]$ generated by Brocot sequences, Mat. Sb. 198 (2007), no. 5, 67-94; English transl., Sb. Math. 198 (2007), no. 5-6, 661-690. MR 2354287 (2008m:11030)

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Additional Information

M. Yu. Vodolagin
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, 1 Leninskiye Gory, GSP\nobreakdash-1, 119991, Moscow, Russia
Email: karganak@gmail.com

DOI: https://doi.org/10.1090/S1061-0022-2014-01305-2
Keywords: M\"onkemeyer algorithm, generalized Farey--Brocot algorithm, multidimensional continued fractions
Received by editor(s): May 27, 2010
Published electronically: June 5, 2014
Additional Notes: The author was supported by RFBR (grant no. 12-01-00681a)
Article copyright: © Copyright 2014 American Mathematical Society

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