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.

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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\textit{O. N. German}, Dokl. Math. 73, No. 1, 38--41 (2006; Zbl 1155.11332); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 406, No. 3, 298--302 (2006)

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##### References:

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