Geodesic
The average number of vertexes of Klein polyhedrons for integer lattices
Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 48-55 
A. A. Illarionov; D. Slinkin. The average number of vertexes of Klein polyhedrons for integer lattices. Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 48-55. http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a4/
@article{DVMG_2011_11_1_a4,
     author = {A. A. Illarionov and D. Slinkin},
     title = {The average number of vertexes of {Klein} polyhedrons for integer lattices},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {48--55},
     year = {2011},
     volume = {11},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a4/}
}
TY  - JOUR
AU  - A. A. Illarionov
AU  - D. Slinkin
TI  - The average number of vertexes of Klein polyhedrons for integer lattices
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2011
SP  - 48
EP  - 55
VL  - 11
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a4/
LA  - ru
ID  - DVMG_2011_11_1_a4
ER  - 
%0 Journal Article
%A A. A. Illarionov
%A D. Slinkin
%T The average number of vertexes of Klein polyhedrons for integer lattices
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2011
%P 48-55
%V 11
%N 1
%U http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a4/
%G ru
%F DVMG_2011_11_1_a4

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

Low estimate for the average number for vertices of Klein polyhedron of integer lattices with given determinant is derived. The low estimate coincides with the high estimate up to a constant. The constant depends on dimension of lattices. High-low estimates for the number of relative minima of integer lattices with given determinant is derived from this fact.

[1] G. F. Voronoi, Sobranie sochinenii v 3-kh tomakh, AN USSR, Kiev, 1952

[2] H. Minkowski, “Generalisation de la theorie des fraction continues”, Ann. Sei. Ecole Norm. Sup., 1896, no. 2, 41–60 | MR | Zbl

[3] F. Klein, “Ueber eine geometrische Auffassung der gewöhlichen Kettenbruchentwichlung”, Nachr. Ges. Wiss. Göttingem., 1895, no. 3, 357–359 | Zbl

[4] V. A. Bykovskii, “Otnositelnye minimumy reshetok i vershiny mnogogrannikov Kleina”, Funkts. analiz i ego pril., 40:1 (2006), 69–71 | DOI | MR | Zbl

[5] V. I. Arnold, Tsepnye drobi, MTsNMO, M., 2001

[6] V. I. Arnold, “Higher dimensional continued fractions”, Nachr. Ges. Wiss. Gottingem., 1998, no. 3, 10–17 | MR | Zbl

[7] V. A. Bykovskii, “O pogreshnosti teoretiko-chislovykh kvadraturnykh formul”, Chebyshevskii sbornik, 3:2 (2002), 27–33 | MR | Zbl

[8] H. Heilbronn, “On the average length of a class of finite continued fractions, Number Theory and Analysis”, Number Theory and Analysis (Papers in Honor of Edmund Landau), Plenum, New York, 1969, 87–96 | DOI | MR

[9] A. A. Illarionov, “Srednee kolichestvo otnositelnykh minimumov trekhmernykh tselochislennykh reshetok”, Algebra i analiz, 23 (2011) (to appear) | MR

[10] A. A. Illarionov, “Statisticheskie svoistva mnogomernykh analogov nepreryvnykh drobei”, Materialy XXII kraevogo konkursa molodykh uchenykh, Izd-vo Tikhookean. gos. un-ta, 2010, 5–16

[11] M. O. Avdeeva, “O nizhnikh otsenkakh kolichestva lokalnykh minimumov tselochislennykh reshëtok”, Fundament. i prikl. matem., 11:6 (2005), 9–14 | MR

[12] A. A. Illarionov, “Otsenki kolichestva otnositelnykh minimumov nepolnykh tselochislennykh reshetok proizvolnogo ranga”, DAN, 418:2 (2008), 155–168 | MR | Zbl

[13] Dzh. Kassels, Geometriya chisel, Mir, M., 1965, 211 pp. | MR

[14] L. Dauner,B. Gryunbaum,V. Kli, Teorema Khelli, Mir, M., 1968, 160 pp.