Discrete analogues of John's theorem

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 $P\left(A,b\right)\subset {\mathbb{Z}}^n$ 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 $P\left(A,b\right)\subset K\cap {\mathbb{Z}}^n\subset P\left(A,O{\left(n\right)}^{3n∕2}b\right)$. 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.