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Estimates on conjectures of Minkowski and woods II

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Abstract

Let ℝn be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < Rwhich contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ℝn of radius \(\sqrt n /2\) contains a point of ∧. This is known to be true for n ≤ 8. Recently we gave estimates on a more general conjecture of Woods for n ≥ 9. This lead to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. Here we shall refine our method to obtain improved estimates for Woods Conjecture. These give improved estimates of Minkowski’s conjecture for 9 ≤ n ≤ 31.

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Authors and Affiliations

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

    R. J. Hans-Gill & Madhu Raka

  2. Department of Applied Sciences, PEC University of Technology, Chandigarh, 160012, India

    Ranjeet Sehmi

Authors
  1. R. J. Hans-Gill
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  2. Madhu Raka
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  3. Ranjeet Sehmi
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Correspondence to R. J. Hans-Gill.

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Dedicated to Professor R. P. Bambah on his 85th Birthday

The author acknowledges the support of Indian National Science Academy as INSA Senior Scientist during the period of preparation of this paper.

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Hans-Gill, R.J., Raka, M. & Sehmi, R. Estimates on conjectures of Minkowski and woods II. Indian J Pure Appl Math 42, 307–333 (2011). https://doi.org/10.1007/s13226-011-0021-9

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  • DOI: https://doi.org/10.1007/s13226-011-0021-9

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