Sharp bounds for a discrete John’s theorem

Abstract

Abstract Tao and Vu showed that every centrally symmetric convex progression $C\subset \mathbb{Z}^d$ is contained in a generalized arithmetic progression of size $d^{O(d^2)} \# C$ . Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$ . We obtain the bound $d^{O(d)} \# C$ , which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John’s theorem.