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.
"Discrete analogues of John's theorem." Mosc. J. Comb. Number Theory 8 (4) 367 - 378, 2019. https://doi.org/10.2140/moscow.2019.8.367