  As a discrete counterpart to the classical John theorem on the approximation
of (symmetric) $n$-dimensional convex bodies $K$ by ellipsoids, Tao and Vu
introduced so called generalized arithmetic progressions $P(A,b)\subset Z^n$ 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 $P(A,b)$ such that $P(A,b)\subset K\cap Z^n\subset
P(A,O(n)^{3n/2}b)$. Here we show that this bound can be lowered to $n^{O(\ln
n)}$ and study some general properties of so called unimodular generalized
arithmetic progressions.
