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On conjectures of Minkowski and Woods for [formula omitted]

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
129
Issue
5
Identifiers
DOI: 10.1016/j.jnt.2008.10.020
Keywords
  • Lattice
  • Covering
  • Non-Homogeneous
  • Product Of Linear Forms
  • Critical Determinant
Disciplines
  • Mathematics

Abstract

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 .

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