Far Eastern Mathematical Journal

To content of the issue


The average length of Minkowski's diagonal continued fractions


O. A. Gorkusha

2011, issue 1, Ñ. 10–27


Abstract
We prove asymptotic formulae with two significant terms for the expectation of the random variable $s(c/d;1)$ — length of Minkowski's diagonal continued fraction when the variables $c$ and $d$ range over the set $1 \le c \le d \le R<\infty$.

Keywords:
Continued fractions, geometry of numbers, lattices

Download the article (PDF-file)

References

[1] H. Minkowski, “Zur Theorie der Kettenbruche”, Annales de l'Ecole Normale Superieure, 13:3 (1894), 41–60
[2] B. Valle?e, “Dynamical analysis of a class of Euclidean Algorithms”, Theoret. Comput. Sci., 297 (2003), 447–486
[3] Y. Hartono, Ergotic properties of continued fraction algorithms, Thesis (Dr.)–Technische Universiteit Delft (The Netherlands), 2003, ISBN: 90-407-2381-8, 119 s.
[4] J. W. Porter, “On a theorem of Heilbronn”, Mathematika, 22:1 (1975), 20–28
[5] D. E. Knuth, “Evaluation of Porter's Constant”, Comp. and Maths. with Appls., 2 (1976), 137–139
[6] G. H. Norton, “On the asymptotic analysis of the Euclidean algorithm”, J. Symbolic Comput., 10:1 (1990), 53–58
[7] A. V. Ustinov, “Asimptoticheskoe povedenie pervogo i vtorogo momenta dlja chisla shagov v algoritme Jevklida”, Izvestija RAN, 72:5 (2008), 189–224
[8] V. Baladi, B. Valle?e, “Euclidean algorithms are Gaussian”, J. Number Theory, 110 (2005), 331–386
[9] A. V. Ustinov, “O srednem chisle shagov v algoritme Evklida s vyborom minimal'nogo po modulju ostatka”, Matematicheskie zametki, 85:1 (2009), 153–156
[10].A. V. Ustinov, “O srednem chisle shagov v algoritme Evklida s nechetnymi nepolnymi chastnymi”, Matematicheskie zametki, 88:4 (2010), 594–604
[11] O.A. Gorkusha, “O konechnyh cepnyh drobjah special'nogo vida”, Chebyshevskij sbornik, 9:1(25) (2008), 80–108
[12] A. A. Karacuba, Osnovy analiticheskoj teorii chisel, Nauka, M., 1983, 500 s.
[13] A. Ja. Hinchin, Izbrannye trudy po teorii chisel, MCNMO, M., 2006, 260 s.
[14] D. E. Knuth, A. C. Yao, “Analysis of the Subtractive Algorithm for Greatest Common Divisors”, Proc. Nat. Acad. Sci. USA, 72:12 (1975), 4720–4722

To content of the issue