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

TL;DR

This paper establishes sharp bounds for the size of generalized arithmetic progressions containing symmetric convex progressions in integer lattices, improving previous bounds and aligning with continuous analogs.

Contribution

It introduces a new bound of d^{O(d)} for the size of generalized arithmetic progressions, sharpening previous results and matching the continuous case.

Findings

New bound d^{O(d)} matches the continuous setting.

Improves previous bounds from d^{O(d^2)} and d^{O(d log d)}.

Bound is sharp up to constants.

Abstract

Tao and Vu showed that every centrally symmetric convex progression $C\subset\mathbb{Z}^d$ is contained in a generalised 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.