Refined Estimates on Conjectures of Woods and Minkowski

TL;DR

This paper improves estimates related to Woods' conjecture in Geometry of Numbers for dimensions 10 to 33, which in turn advances understanding of Minkowski's classical conjecture on linear forms.

Contribution

The paper provides new bounds on Woods' conjecture for dimensions 10 to 33, extending known results and refining the classical Minkowski conjecture estimates.

Findings

Improved bounds for Woods' conjecture in dimensions 10-33.

Enhanced estimates for Minkowski's conjecture on linear forms.

Extension of validity of Woods' conjecture beyond dimension 9.

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.