Geodesic
The average number of relative minima of three-dimensional integer lattices of a~given determinant
Izvestiya. Mathematics , Tome 76 (2012) no. 3, pp. 535-562.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain an asymptotic formula for the average number of relative minima of the three-dimensional complete integer lattices of a given determinant. This generalizes Heilbronn's classical result on the average length of a finite continued fraction with a fixed denominator.
Keywords: relative minimum, multidimensional continued fraction, average length of continued fractions. 
@article{IM2_2012_76_3_a4,
     author = {A. A. Illarionov},
     title = {The average number of relative minima of three-dimensional integer lattices of a~given determinant},
     journal = {Izvestiya. Mathematics },
     pages = {535--562},
     publisher = {mathdoc},
     volume = {76},
     number = {3},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a4/}
}
TY  - JOUR
AU  - A. A. Illarionov
TI  - The average number of relative minima of three-dimensional integer lattices of a~given determinant
JO  - Izvestiya. Mathematics 
PY  - 2012
SP  - 535
EP  - 562
VL  - 76
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a4/
LA  - en
ID  - IM2_2012_76_3_a4
ER  - 
%0 Journal Article
%A A. A. Illarionov
%T The average number of relative minima of three-dimensional integer lattices of a~given determinant
%J Izvestiya. Mathematics 
%D 2012
%P 535-562
%V 76
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a4/
%G en
%F IM2_2012_76_3_a4
A. A. Illarionov. The average number of relative minima of three-dimensional integer lattices of a~given determinant. Izvestiya. Mathematics , Tome 76 (2012) no. 3, pp. 535-562. http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a4/

[1] G. F. Voronoi, Sobranie sochinenii, v. 1, Izd-vo AN USSR, Kiev, 1952 | MR | Zbl

[2] H. Minkowski, “Généralisation de la théorie des fractions continues”, Ann. Sci. École Norm. Sup. (3), 13 (1896), 41–60 | MR | Zbl

[3] V. A. Bykovskii, “On the error of number-theoretic quadrature formulas”, Dokl. Math., 67:2 (2003), 175–176 | MR | Zbl

[4] N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, MTsNMO, M., 2004 | MR | Zbl

[5] O. A. Gorkusha, N. M. Dobrovolskii, “Ob otsenkakh giperbolicheskoi dzeta-funktsii reshetok”, Chebyshevskii sb., 6:2 (2005), 129–137 | MR | Zbl

[6] M. O. Avdeeva, V. A. Bykovskii, “Upper and lower bounds for the Voronoi–Minkowski constant”, Math. Notes, 87:3–4 (2010), 457–465 | DOI | MR | Zbl

[7] A. A. Illarionov, “Estimate for the number of relative minima of arbitrary-rank incomplete integer lattices”, Dokl. Math., 77:1 (2008), 31–34 | DOI | MR | Zbl

[8] H. Heilbronn, “On the average length of a class of finite continued fractions”, Number theory and analysis, Plenum, New York, 1969, 87–96 | MR | Zbl

[9] A. A. Illarionov, “Srednee kolichestvo otnositelnykh minimumov trekhmernykh tselochislennykh reshetok”, Algebra i analiz, 23:3 (2011), 189–215

[10] A. A. Illarionov, “Statisticheskie svoistva mnogomernykh analogov nepreryvnykh drobei”, Nauka – Khabarovskomu krayu: materialy XII kraevogo konkursa molodykh uchenykh, Izd-vo Tikhookean. gos. un-ta, Khabarovsk, 2010, 5–15

[11] J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | MR | Zbl | Zbl

[12] P. G. L. Dirichlet, Vorlezungen über Zahlentheorie, Amer. Math. Soc., Providence, RI, 1894 | MR | Zbl

[13] A. V. Ustinov, “The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments”, Sb. Math., 200:4 (2009), 597–627 | DOI | MR | Zbl