Discrete & Computational Geometry

, Volume 35, Issue 3, pp 429–435 | Cite as

Lattice Points in Large Borel Sets and Successive Minima

  • Iskander AlievEmail author
  • Peter M. GruberEmail author


Let B be a Borel set in Ed with volume V(B) = ∞. It is shown that almost all lattices L in Ed contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in Ed with V(S ) = ∞. Then for almost all lattices L in Ed the successive minima λ1(S,L),..., λd(S,L) of S with respect to L are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima.


Computational Mathematic Volume Versus Lattice Point Independent Point Baire Category 
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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZScotland
  2. 2.Forschungsgruppe, Konvexe und Diskrete Geometrie, Technische Universitat Wien, Wiedner Hauptstrass e 8-10/1046, A-1040 ViennaAustria

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