On conjectures of minkowski and woods for n=7

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


Let Rnbe 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 Rn 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 R7. Together with a result of C.T. McMullen (2005), the long standing conjecture of Minkowski follows for n=7.

Item Type:Article
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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