Refined Estimates on Conjectures of Woods and Minkowski

Let Λ be a lattice in Rⁿ reduced in the sense of Korkine and Zolotareff having a basis of the form (A₁, 0, 0, . . . , 0), (a₂,1,A₂, 0, . . . , 0), . . . , (a_n,1, an,2, . . . , a_n,n−1,A_n) where A₁,A₂, . . . ,A_n are all positive. A well known conjecture of Woods in Geometry of Numbers asserts that if A₁A₂· · ·A_n = 1 and A_i ≤ A₁ for each i then any closed sphere in Rⁿ of radius √n/2 contains a point of Λ. Woods' Conjecture is known to be true for n ≤ 9. In this paper we give estimates on the Conjecture of Woods for 10 ≤ n ≤ 33, improving the earlier best known results of Hans-Gill et al. These lead to an improvement, for these values of n, to the estimates on the long standing classical conjecture of Minkowski on the product of n non-homogeneous linear forms.