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Klein polyhedra and norm minima of lattices. (English. Russian original) Zbl 1155.11332
Dokl. Math. 73, No. 1, 38-41 (2006); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 406, No. 3, 298-302 (2006).
Summary: In this paper, we investigate one of the most natural multidimensional geometric generalizations of continued fractions, the so-called Klein polyhedra. The main result of this paper is the multidimensional generalization of the well-known statement that a number is badly approximable if and only if its partial quotients are bounded. Recall that a number \(\alpha\) is said to be badly approximable if there exists a constant \(c>0\) such that, for all integers \(p\) and positive integers \(q\), the following inequality holds: \[ |q\alpha-p|\geq \frac cq. \] Since as a multidimensional generalization of continued fractions we consider Klein polygons, it is natural to consider the property of a lattice \(\Lambda\) to have positive norm minimum as a multidimensional analogue of the property of a number to be badly approximable.
Definition. The norm minimum of a lattice \(\Lambda\) is defined as \[ N(\Lambda)=\inf_{x\in\Lambda\setminus\{0\}}|\varphi (x)|, \] where \(\varphi(x)=x_1x_2\dots x_n\). Theorem. Let \(\Lambda \subset\mathbb{R}^n\) be an \(n\)-dimensional irrational lattice. Then, \(N (\Lambda)>0\) if and only if the faces and the edge stars of vertices of the sail generated by \(\Lambda\) have uniformly bounded determinants.
Section 3 gives the relation to the Littlewood and Oppenheim conjectures, Section 4 treats uniform boundedness of determinants of sail faces, Section 5, dual lattices and polar polyhedra, Section 6 boundedness away from zero of the form \(\varphi(x)\) in the positive orthant, and Section 7, the logarithmic plane.

MSC:
11H50 Minima of forms
11H06 Lattices and convex bodies (number-theoretic aspects)
11H46 Products of linear forms
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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