Vol. 8, No. 4, 2019

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Discrete analogues of John's theorem

Sören Lennart Berg and Martin Henk

Vol. 8 (2019), No. 4, 367–378

As a discrete counterpart to the classical theorem of Fritz John on the approximation of symmetric n-dimensional convex bodies K by ellipsoids, Tao and Vu introduced so called generalized arithmetic progressions P(A,b) n in order to cover (many of) the lattice points inside a convex body by a simple geometric structure. Among others, they proved that there exists a generalized arithmetic progressions P(A,b) such that P(A,b) K n P(A,O(n)3n2b). Here we show that this bound can be lowered to nO(ln n) and study some general properties of so called unimodular generalized arithmetic progressions.

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John's theorem, arithmetic progressions, convex bodies, lattices
Mathematical Subject Classification 2010
Primary: 11H06, 52C07
Secondary: 52A40
Received: 14 April 2019
Revised: 27 May 2019
Accepted: 16 June 2019
Published: 11 October 2019
Sören Lennart Berg
Institut für Mathematik
Technische Universität Berlin
Martin Henk
Institut für Mathematik
Technische Universität Berlin