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Abstract
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As a discrete counterpart to the classical theorem of Fritz John on the approximation of symmetric
-dimensional
convex bodies
by ellipsoids, Tao and Vu introduced so called generalized arithmetic progressions
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
such that
. Here we show that this
bound can be lowered to
and study some general properties of so called unimodular generalized arithmetic
progressions.
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Keywords
John's theorem, arithmetic progressions, convex bodies,
lattices
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Mathematical Subject Classification 2010
Primary: 11H06, 52C07
Secondary: 52A40
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Milestones
Received: 14 April 2019
Revised: 27 May 2019
Accepted: 16 June 2019
Published: 11 October 2019
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