Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

 
 

 

Average number of local minima for three-dimensional integral lattices


Author: A. A. Illarionov
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 551-570
MSC (2010): Primary 11H06
DOI: https://doi.org/10.1090/S1061-0022-2012-01208-2
Published electronically: March 2, 2012
MathSciNet review: 2896168
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An asymptotic formula is found for the average number of local minima of three-dimensional complete integral lattices with determinant in the interval $ [1,N]$. This is a generalization to the two-dimensional case of the classical result about the average length of a finite continued fraction with denominator belonging to $ [1,N]$.


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

  • 1. F. Klein, Uber eine geometrische Auffassung der gewohlichen Kettenbruchentwichlung, Nachr. Ges. Wiss. Göttingen 1895, no. 3, 357-359.
  • 2. G. F. Voronoĭ, Collected works in three volumes, Akad. Nauk USSR, Kiev, 1952. (Russian) MR 0062686 (16:2d)
  • 3. H. Minkowski, Généralisation de la théorie des fractions continues, Ann. Sci. École Norm. Sup. (3) 13 (1896), 41-60. MR 1508923
  • 4. V. A. Bykovskiĭ, On the error of number-theoretic quadrature formulas, Dokl. Akad. Nauk 389 (2003), no. 2, 154-155. (Russian) MR 2003229
  • 5. -, On the error in number-theoretic quadrature formulas, Chebyshevskiĭ Sb. 3 (2002), no. 2, 27-33. (Russian) MR 2035611 (2005d:11102)
  • 6. N. M. Korobov, Number-theoretic methods in approximate analysis, 2nd ed., MTsNMO, Moscow, 2004. (Russian) MR 2078157 (2005f:41001)
  • 7. O. A. Gorkusha and N. M. Dobrovol'skiĭ, On estimates for the hyperbolic zeta function of lattices, Chebyshevskiĭ Sb. 6 (2005), no. 2, 129-137. (Russian) MR 2262602 (2007j:11118)
  • 8. A. A. Illarionov, An estimate for the number of relative minima of arbitrary-rank incomplete integer lattices, Dokl. Akad. Nauk 418 (2008), no. 8, 155-158; English transl., Dokl. Math. 77 (2008), no. 1, 31-34. MR 2462063
  • 9. H. Heilbronn, On the average length of a class of finite continued fractions, Number Theory and Analysis, Papers in Honor of Edmund Landau, Plenum, New York, 1969, pp. 87-96. MR 0258760 (41:3406)
  • 10. M. O. Avdeeva, Lower bounds for the number of local minima of integer lattices, Fundam. Prikl. Mat. 11 (2005), no. 6, 9-14; English transl., J. Math. Sci. (N. Y.) 146 (2007), no. 2, 5629-5633. MR 2204418 (2006k:11127)
  • 11. G. Lochs, Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmässigen Kettenbrüche., Monatsh. Math. 65 (1961), 27-52. MR 0124308 (23:A1622)
  • 12. J. W. Porter, On a theorem of Heilbronn, Mathematika 22 (1975), no. 1, 20-28. MR 0498452 (58:16567)
  • 13. D. E. Knuth, The art of computer programming. Vol. 2, Addison-Wesley Publ. Co., Reading, MA, 1981. MR 0633878 (83i:68003)
  • 14. H. Minkowski, Zur Theorie der Kettenbrüche, Gesammelte Abhandlungen. Bd. I, Teubner, Leipzig-Berlin, 1911, S. 278-292.
  • 15. V. A. Bykovskiĭ and O. A. Gorkusha, Minimal bases of three-dimensional complete lattices, Mat. Sb. 192 (2001), no. 2, 57-66; English transl., Sb. Math. 192 (2001), no. 1-2, 215-223. MR 1835985 (2002d:11075)
  • 16. O. A. Gorkusha, Minimal bases of three-dimensional complete lattices, Mat. Zametki 69 (2001), no. 3, 353-362; English transl., Math. Notes 69 (2001), no. 3-4, 320-328. MR 1846834 (2002d:11077)
  • 17. J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin, 1997. MR 1434478 (97i:11074)
  • 18. A. V. Ustinov, Solution of the Arnol'd problem on weak asymptotics for Frobenius numbers with three arguments, Mat. Sb. 200 (2009), no. 4, 131-160; English transl., Sb. Math. 200 (2009), no. 3-4, 597-627. MR 2531884 (2010e:11091)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 11H06

Retrieve articles in all journals with MSC (2010): 11H06


Additional Information

A. A. Illarionov
Affiliation: Khabarovsk Division, Institute of Applied Mathematics, Russian Academy of Sciences, 54 Dzerzhinskiǐ Street, Khabarovsk 680000, Russia
Email: illar_a@list.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01208-2
Keywords: Lattice, local minimum, multidimensional continued fraction, Euclid algorithm
Received by editor(s): November 30, 2009
Published electronically: March 2, 2012
Additional Notes: Supported by RFBR (grants nos. 10-01-98002r-siberia-a , 11-01-00628-a), by FED RAS (grants nos. 11-III-V-01M-002, 09-I-114-03), and by the grant MD-2339.2010.1 of the President of RF
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society