Refined Estimates on Conjectures of Woods and Minkowski-I

[摘要] Let ^ be a lattice in Rn reduced in the sense of Korkine and Zolotare having a basis of the form (A1, 0, 0, . . . , 0),(a2,1,A2, 0, . . . , 0), . . . , (an,1, an,2, . . . , an,n-1,An) where A1,A2, . . . ,An are all positive. A well known onjecture of Woodsin Geometry of Numbers asserts that if A1 A2…An=1 and Ai £ A1 for each i then any closed sphere in Rn of radiusn / 2 contains a point of ^. Woods' Conjecture is known to be true for n £ 9 . In this paper we obtain estimates onthe Conjecture of Woods for n=10; 11 and 12 improving the earlier best known results of Hans-Gill et al. These leadto an improvement, for these values of n, to the estimates on the long standing classical conjecture of Minkowski onthe product of n non-homogeneous linear forms.

[关键词] Lattice;Covering;Non-homogeneous;Product of linearforms;Critical determinant;Korkine and Zolotare reduction;Hermite'sconstant;Centre density