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Cylindrical minima of integral lattices. (English. Russian original) Zbl 1273.11103
St. Petersbg. Math. J. 24, No. 2, 301-312 (2013); translation from Algebra Anal. 24, No. 2, 154-170 (2012).
Summary: Let \(\Phi\) be a norm in \(\mathbb R^{s-1}\). A nonzero node \(\gamma =(\gamma _1,\dots ,\gamma _s)\) of an \(s\)-dimensional lattice \(\Gamma\) is called a \(\Phi\)-cylindrical minimum of \(\Gamma\) if there is no other nonzero node \(\eta =(\eta _1,\dots ,\eta _s)\) with \[ \Phi (\gamma _1,\dots ,\gamma _{s-1})\leq\Phi (\eta _1,\dots ,\eta _{s-1}), \quad |\eta _s| \leq |\gamma _s|, \] where at least one inequality is strict. It is proved that the average number of the \(\Phi\)-cylindrical minima of \(s\)-dimensional integer lattices whose determinant belongs to \([1;N]\) is equal to \(\mathcal{C}_s(\Phi )\cdot\ln N + O_{s,\Phi }(1)\), where \(\mathcal {C}_s(\Phi )\) is a positive constant depending only on \(s\) and \(\Phi\). This formula is a version of the classical result about the average length of a finite continued fraction.

MSC:
11H06 Lattices and convex bodies (number-theoretic aspects)
11H50 Minima of forms
11J70 Continued fractions and generalizations
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[1] G. F. Voronoĭ, Sobranie sočineniĭ v treh tomah, Izdatel\(^{\prime}\)stvo Akademii Nauk Ukrainskoĭ SSR, Kiev., 1952 1953 (Russian).
[2] Gustav Lochs, Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmässigen Kettenbrüche, Monatsh. Math. 65 (1961), 27 – 52 (German). · Zbl 0097.03602
[3] H. Heilbronn, On the average length of a class of finite continued fractions, Number Theory and Analysis (Papers in Honor of Edmund Landau), Plenum, New York, 1969, pp. 87 – 96.
[4] A. A. Illarionov, The average number of relative minima of three-dimensional integer lattices, Algebra i Analiz 23 (2011), no. 3, 189 – 215 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 3, 551 – 570.
[5] A. A. Illarionov, On the cylindrical minima of three-dimensional lattices, Dal\(^{\prime}\)nevost. Mat. Zh. 11 (2011), no. 1, 37 – 47 (Russian, with English and Russian summaries). · Zbl 1261.11049
[6] B. N. Delone and D. K. Faddeev, Theory of Irrationalities of Third Degree, Acad. Sci. URSS. Trav. Inst. Math. Stekloff, 11 (1940), 340 (Russian). B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. · Zbl 0061.09001
[7] J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. · Zbl 0866.11041
[8] J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. · Zbl 0395.10029
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