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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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##### References:
 [1] P. Erdös, P. M. Gruber, and J. Hammer, Lattice Points (Longman, Harlow, 1989). · Zbl 0683.10025 [2] O. N. German, Mat. Sb. 196(3), 31–60 (2005). [3] J.-O. Moussafir, Zap. Nauchn. Sem. St. Petersburg. Otdel. Mat. Inst. Steklov (POMI) 256 (2000). [4] J. W. S. Cassles and H. P. F. Swinnerton-Dyer, Philos. Trans. Roy. Soc. (London) A248, 73–96 (1955). · Zbl 0065.27905 [5] B. F. Skubenko, Zap. Nauchn. Sem. LOMI 168 (1988). [6] B. F. Skubenko, Zap. Nauchn. Sem. LOMI 183 (1990).
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