FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
The average number of vertexes of Klein polyhedrons for integer lattices |
A. A. Illarionov, D. Slinkin |
2011, issue 1, P. 48–55 |
Abstract |
| 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. |
Keywords: high dimension continued fraction, relative minimum, Klein polyhedron |
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References |
| [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, № 2, 41–60. [3] F. Klein, “Ueber eine geometrische Auffassung der gewo?hlichen Kettenbruchentwichlung”, Nachr. Ges. Wiss. Go?ttingem., 1895, № 3, 357–359. [4] V. A. Bykovskii, “Otnositel'nye minimumy reshetok i vershiny mnogogrannikov Kleina”, Funkts. analiz i ego pril., 40:1 (2006), 69–71. [5] V. I. Arnol'd, Tsepnye drobi, MTsNMO, M., 2001. [6] V. I. Arnold, “Higher dimensional continued fractions”, Nachr. Ges. Wiss. Gottingem., 1998, № 3, 10–17. [7] V. A. Bykovskii, “O pogreshnosti teoretiko-chislovykh kvadraturnykh formul”, Chebyshevskii sbornik, 3:2 (2002), 27–33. [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. [9] A. A. Illarionov, “Srednee kolichestvo otnositel'nykh minimumov trekhmernykh tselochislennykh reshetok”, Algebra i analiz, 23 (2011) (v pechati). [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 lokal'nykh minimumov tselochislennykh reshe?tok”, Fundament. i prikl. matem., 11:6 (2005), 9–14. [12] A. A. Illarionov, “Otsenki kolichestva otnositel'nykh minimumov nepolnykh tselochislennykh reshetok proizvol'nogo ranga”, DAN, 418:2 (2008), 155–168. [13] Dzh. Kassels, Geometriia chisel, Mir, M., 1965, 211 s. [14] L. Dauner,B. Griunbaum,V. Kli, Teorema Khelli, Mir, M., 1968, 160 s. |