Covering lattice points by subspaces

Bárány, Imre and Harcos, Gergely and Pach, János and Tardos, Gábor (2002) Covering lattice points by subspaces. Periodica Mathematica Hungarica, 43 (1-2). pp. 93-103. ISSN 0031-5303 (print), 1588-2829 (online)

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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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