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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.

11H06 Lattices and convex bodies (number-theoretic aspects)
11H50 Minima of forms
11J70 Continued fractions and generalizations
Full Text: DOI
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