Covering lattice points by subspaces |
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Authors: | Bárány Imre Harcos Gergely Pach János Tardos Gábor |
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Affiliation: | (1) Rényi Institute, Hungarian Academy of Sciences, Budapest, POB 127, H-1364, Hungary;(2) Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA;(3) Rényi Institute, Hungarian Academy of Sciences, Budapest, POB 127, H-1364, Hungary |
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Abstract: | We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body C , symmetric about the origin 0. This enables us to prove the following statement, which settles a problem of G. Halász. The maximum number of n-wise linearly independent lattice points in the n-dimensional ball r B n of radius r around 0 is O(rn/(n-1)). This bound cannot be improved. We also show that the order of magnitude of the number of diferent (n - 1)-dimensional subspaces induced by the lattice points in r&Bgr;n is rn/(n-1). |
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