Improved bounds for a discrete John-type theorem

Tao and Vu introduced a discrete analogue of John’s theorem in which convex progressions are approximated by generalized arithmetic progressions. In the covering version of this problem, one asks for a small GAP containing all lattice points of a given origin-symmetric convex body. We prove that every such convex progression in dimension $n$ admits an infinitely proper GAP cover whose size is within a factor $O(n{)}^{2n}$ of the cardinality of the original set, improving the previously known factor $O(n{)}^{3n}$. We also show that a loss of order $\mathrm{\Omega }(n{)}^n$ is unavoidable for infinitely proper GAP covers.