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Successive Minima and Best Simultaneous Diophantine Approximations
We study the problem of best approximations of a vector by rational vectors of a lattice whose
common denominator is bounded. To this end we introduce successive minima for a
periodic lattice structure and extend some classical results from geometry of
numbers to this structure. This leads to bounds for the best approximation
problem which generalize and improve former results.Comment: 8 page
Mahler's work on the geometry of numbers
Mahler has written many papers on the geometry of numbers. Arguably, his most
influential achievements in this area are his compactness theorem for lattices,
his work on star bodies and their critical lattices, and his estimates for the
successive minima of reciprocal convex bodies and compound convex bodies. We
give a, by far not complete, overview of Mahler's work on these topics and
their impact.Comment: 17 pages. This paper will appear in "Mahler Selecta", a volume
dedicated to the work of Kurt Mahler and its impac
Notes on lattice points of zonotopes and lattice-face polytopes
Minkowski's second theorem on successive minima gives an upper bound on the
volume of a convex body in terms of its successive minima. We study the problem
to generalize Minkowski's bound by replacing the volume by the lattice point
enumerator of a convex body. In this context we are interested in bounds on the
coefficients of Ehrhart polynomials of lattice polytopes via the successive
minima. Our results for lattice zonotopes and lattice-face polytopes imply, in
particular, that for 0-symmetric lattice-face polytopes and lattice
parallelepipeds the volume can be replaced by the lattice point enumerator.Comment: 16 pages, incorporated referee remarks, corrected proof of Theorem
1.2, added new co-autho
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