Lattice-point enumerators of ellipsoids

Abstract

Minkowski’s second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice Λ can be bounded above by a quantity involving all the successive minima of K with respect to Λ. We will prove here that the number of lattice points inside K can also accept an upper bound of roughly the same size, in the special case where K is an ellipsoid. Whether this is also true for all K unconditionally is an open problem, but there is reasonable hope that the inductive approach used for ellipsoids could be extended to all cases.

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Correspondence to Romanos-Diogenes Malikiosis.

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Malikiosis, RD. Lattice-point enumerators of ellipsoids. Combinatorica 33, 733–744 (2013). https://doi.org/10.1007/s00493-013-2943-9

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Mathematics Subject Classification (2000)

  • 11H06
  • 52C07