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.

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