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Refined Estimates on Conjectures of Woods and Minkowski


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1 Centre for Advanced Study in Mathematics Panjab University, Chandigarh-160014, India
     

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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), . . . , (an,1, an,2, . . . , an,n−1,An) where A1,A2, . . . ,An are all positive. A well known conjecture of Woods in Geometry of Numbers asserts that if A1A2· · ·An = 1 and Ai ≤ 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.

Keywords

Lattice, Covering, Non-Homogeneous, Product of Linear Forms, Critical Determinant, Korkine and Zolotareff Reduction, Hermite’s Constant, Center Density.
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  • Refined Estimates on Conjectures of Woods and Minkowski

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Authors

Leetika Kathuria
Centre for Advanced Study in Mathematics Panjab University, Chandigarh-160014, India
Madhu Raka
Centre for Advanced Study in Mathematics Panjab University, Chandigarh-160014, India

Abstract


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), . . . , (an,1, an,2, . . . , an,n−1,An) where A1,A2, . . . ,An are all positive. A well known conjecture of Woods in Geometry of Numbers asserts that if A1A2· · ·An = 1 and Ai ≤ 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.

Keywords


Lattice, Covering, Non-Homogeneous, Product of Linear Forms, Critical Determinant, Korkine and Zolotareff Reduction, Hermite’s Constant, Center Density.

References