Refined Estimates on Conjectures of Woods and Minkowski

Abstract

Let $\wedge$ be a lattice in $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff having a basis of the form $(A_1,0,0,\ldots,0),(a_{2,1},A_2,0,\ldots,0)$, $\ldots,(a_{n,1},a_{n,2},\ldots,a_{n,n-1},A_n)$ where $A_1, A_2,\ldots,A_n$ are all positive. A well known conjecture of Woods in Geometry of Numbers asserts that if $A_{1}A_{2}\cdots A_{n}=1$ and $A_{i}\leqslant A_{1}$ for each $i$ then any closed sphere in $\mathbb{R}^n $ of radius $ \sqrt{n}/2$ contains a point of $\wedge$. Woods' Conjecture is known to be true for $n\leq 9$. In this paper we give estimates on the Conjecture of Woods for $10\leq n\leq33$, 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.