Lattice-point enumerators of ellipsoids

Details Lattice-point enumerators of ellipsoids Lattice-point enumerators of ellipsoids

2012-02-17

arxiv.org/abs/1202.3876v1

Mathematics

Number Theory

9 pages

Scientific paper

Minkowski's second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded above by a quantity involving all the successive minima of K with respect to \Lambda. 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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