#### 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
##### Abstract

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 $P\left(A,b\right)\subset {ℤ}^{n}$ 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 $P\left(A,b\right)$ such that $P\left(A,b\right)\subset K\cap {ℤ}^{n}\subset P\left(A,O{\left(n\right)}^{3n∕2}b\right)$. Here we show that this bound can be lowered to ${n}^{O\left(lnn\right)}$ and study some general properties of so called unimodular generalized arithmetic progressions.

##### Keywords
John's theorem, arithmetic progressions, convex bodies, lattices
##### Mathematical Subject Classification 2010
Primary: 11H06, 52C07
Secondary: 52A40