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Tao and Vu showed that every centrally symmetric convex progression$C\subset\mathbb{Z}^d$ is contained in a generalised arithmetic progression ofsize $d^{O(d^2)} \# C$. Berg and Henk improved the size bound to $d^{O(d\logd)} \# C$. We obtain the bound $d^{O(d)} \# C$, which is sharp up to theimplied constant, and is of the same form as the bound in the continuoussetting given by John's Theorem.

Sharp bounds for the Tao-Vu Discrete John's Theorem