2015/01/14 by Leetika Kathuria, Kathuria, Leetika, Madhu Raka +1
Mathematics · #11H31 #11H46 #11J20 #11J37 #52C15 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11H31 #msc:11H46 #msc:11J20 #msc:11J37 #msc:52C15
paper · pdf · doi:10.48550/arxiv.1501.03277
63 pages
arxiv created 2015/01/14 · arxiv updated 2015/01/15
Let \wedge 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\leqslant A1 for each i then any closed sphere in ℝn of radius √(n)/2 contains a point of \wedge. 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.