Hans-Gill, R. J. ; Raka, Madhu ; Sehmi, Ranjeet
(2009)
*On conjectures of minkowski and woods for n=7*
Journal of Number Theory, 129
(5).
pp. 1011-1033.
ISSN 0022-314X

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00223...

Related URL: http://dx.doi.org/10.1016/j.jnt.2008.10.020

## Abstract

Let R^{n}be the n-dimensional Euclidean space with O as the origin. Let ∧ be a lattice of determinant 1 such that there is a sphere |X|<R which contains no point of ∧ other than O and has n linearly independent points of ∧ on its boundary. A well-known conjecture in the geometry of numbers asserts that any closed sphere in R^{n} of radius √n/4 contains a point of ∧. This is known to be true for n≤6. Here we prove a more general conjecture of Woods for n=7 from which this conjecture follows in R^{7}. Together with a result of C.T. McMullen (2005), the long standing conjecture of Minkowski follows for n=7.

Item Type: | Article |
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Source: | Copyright of this article belongs to Elsevier Science. |

Keywords: | Lattice; Covering; Non-homogeneous; Product of linear forms; Critical determinant |

ID Code: | 12328 |

Deposited On: | 10 Nov 2010 05:41 |

Last Modified: | 03 Jun 2011 06:26 |

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