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Minimal vector systems in three-dimensional lattices and an analogue of Vahlen’s theorem for three-dimensional Minkowski continued fractions. (English. Russian original) Zbl 1370.11014
Proc. Steklov Inst. Math. 280, Suppl. 2, S91-S116 (2013); translation from Sovrem. Probl. Mat. 16, 103-128 (2012).
From the introduction: In this paper, we suggest an approach to studying minimal systems of vectors in arbitrary lattices. In reducible lattices, the notion of minimality can be given various meanings by imposing more or less strong conditions on the vector system under consideration. Different constraints lead to different three-dimensional analogues of Vahlen’s theorem [K. Th. Vahlen, J. Reine Angew. Math. 115, 221–233 (1895; JFM 26.0230.01)]. In the framework of the approach which we suggest in this paper, for minimal systems of the form used by M. O. Avdeeva and V. A. Bykovskiĭ [Math. Notes 79, No. 2, 151–156 (2006); translation from Mat. Zametki 79, No. 2, 163–168 (2006; Zbl 1135.11035)], a relaxed version \[ a_1a_2a_3 + b_1b_2b_3 + c_1c_2c_3 \leq 2\det\Gamma \] of Vahlen’s theorem is valid, while for completely minimal systems (satisfying more rigid conditions), the sharper estimate \[ a_1a_2a_3 + b_1b_2b_3 + c_1c_2c_3 \leq \det\Gamma \] remains valid. This result is based on a complete classification of minimal systems of vectors in arbitrary three-dimensional lattices.

MSC:
11A55 Continued fractions
11H06 Lattices and convex bodies (number-theoretic aspects)
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[1] F. Klein, ”Über eine geometrische Auffassung der gewöhnlichen Kettenbruchentwickelung,” Gött. Nachr. 3, 357–359 (1895). · JFM 26.0229.02
[2] F. Klein, ”Sur une représentation géométrique du développement en fraction continue ordinaire,” Nouv. Ann. (3) 15, 327–331 (1896).
[3] G. Voronoi, A Generalization of the Continuous Fraction Algorithm, Doctoral Thesis (Warsaw, 1896) [in Russian].
[4] G. Voronoi, Collection of Works (Kiev, 1952), Vol. 1. · Zbl 0347.45021
[5] H. Minkowski, ”Généralisation de la théorie des fractions continues,” Ann. Sci.École Norm. Sup. (3) 13, 41–60 (1896). · JFM 27.0170.01
[6] H. Minkowski, ”Zur Theorie der Kettenbrüche,” in Gesammelte Abhandlungen (Teubner, Leipzig, 1911), Vol 1, pp. 278–292.
[7] V. A. Bykovskii, ”Local minima of lattices and vertices of Klein polyhedra,” Funkts. Anal. Ego Prilozh. 40(1), 69–71 (2006) [Funct. Anal. Appl. 40 (1), 56–57 (2006)]. · Zbl 1152.11031
[8] O. N. German, ”Klein polyhedra and relative minima of lattices,” Mat. Zametki 79(4), 546–552 (2006) [Math. Notes 79 (3–4), 505–510 (2006)]. · Zbl 1119.11036
[9] B. N. Delone and D. K. Faddeev, ”The theory of irrationalities of the third degree,” Tr. Mat. Inst. im. V.A. Steklova Ross. Akad. Nauk 11, 3–340 (1940). · Zbl 0061.09001
[10] B. N. Delone, The St. Petersburg School of Number Theory (Akad. Nauk SSSR, Moscow-Leningrad, 1947) [in Russian]. · Zbl 0033.10403
[11] H. Hancock, Development of the Minkowski Geometry of Numbers (Dover, New York, 1964), Vols. 1, 2. · Zbl 0123.25603
[12] V. A. Bykovskii, ”On the error of number-theoretic quadrature formulas,” Dokl. Ross. Akad. Nauk 389(2), 154–155 (2003) [Dokl. Math. 67 (2), 175–176 (2003)]. · Zbl 1247.65030
[13] V. A. Bykovskii, ”The discrepancy of the Korobov lattice points,” Izv. Ross. Akad. Nauk Ser. Mat. 76(3), 19–38 (2012) [Izv.: Math. 76 (3), 446 (2012)].
[14] N. M. Korobov, Number-Theoretic Methods in Approximate Analysis (MTsNMO, Moscow, 2004) [in Russian]. · Zbl 1143.65024
[15] M. O. Avdeeva and V. A. Bykovskii, ”Refinement of Vahlen’s theorem for Minkowski bases of three-dimensional lattices,” Mat. Zametki 79(2), 163–168 (2006) [Math. Notes 79 (1–2), 151–156 (2006)].
[16] K. Th. Vahlen, ”Über Näherungswerte und Kettenbrüche,” J. Reine Angew.Math. 115(3), 221–233 (1895). · JFM 26.0230.01
[17] A. Ya. Khinchin, Continued Fractions (Nauka, Moscow, 1978) [in Russian].
[18] M. O. Avdeeva and V. A. Bykovskii, ”An analogue of Vahlen’s theorem for simultaneous approximations of a pair of numbers,” Mat. Sb. 194(7), 3–14 (2003) [Sbornik: Math. 194 (7), 955–967 (2003)].
[19] V. A. Bykovskii, ”Vahlen theorem for two-dimensional convergents,” Mat. Zametki 66(1), 30–37 (1999) [Math. Notes 66 (1), 24–29 (1999)]. · Zbl 0981.11024
[20] S. V. Gassan, ”The structure of Vahlen domains for three-dimensional lattices,” Chebyshev Sb. 6(3), 51–84 (2005). · Zbl 1174.11054
[21] A. V. Ustinov, ”To Vahlen’s three-dimensional theorem,” Mat. Zametki (in press). · Zbl 1370.11014
[22] V. A. Bykovskii and O. A. Gorkusha, ”Minimal bases of three-dimensional complete lattices,” Mat. Sb. 192(2), 57–66 (2001) [Sbornik: Math. 192 (2), 215–223 (2001)]. · Zbl 1020.11047
[23] Ph. Furtwängler and M. Zeisel, ”Zur Minkowskischen Parallelepipedapproximation”,Monatsh. Math. Phys. 30(1), 177–198 (1920). · JFM 47.0165.05
[24] P. M. Pepper, ”Une application de la géométrie des nombres à une généralisation d’une fraction continue”, Ann. Sci.École Norm. Sup. (3) 56, 1–70 (1939). · JFM 65.0179.02
[25] H. Minkowski, Diophantische Approximationen. Eine Einführung in die Zahlentheorie (Teubner, Leipzig, 1907).
[26] O. A. Gorkusha, ”Minimal bases in complete 3-lattices,” Mat. Zametki 69(3), 353–362 (2001) [Math. Notes 69 (3–4), 320–328 (2001)]. · Zbl 0997.11048
[27] H. Minkowski, Gesammelte Abhandlungen (Teubner, Leipzig, 1911), Vol. 2.
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