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 $(A_1,0,0,\dots ,0),(a_{2,1},A_2,0,\dots ,0)$, $\dots ,(a_{n,1},a_{n,2},\dots ,a_{n,n-1},A_n)$ where $A_1,A_2,\dots ,A_n$ 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 $A_i⩽A_1$ 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 $n\le 9$. In this paper we give estimates on the Conjecture of Woods for $10\le n\le 33$, 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.

63 pages

References in corpus (1)

• On Conjectures of Minkowski and Woods for n=9