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

2023/09/21 by Peter van Hintum, Peter Keevash, van Hintum, Peter +1
Mathematics · #Limits and Structures in Graph Theory #Advanced Topology and Set Theory #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2309.12386

Abstract

Tao and Vu showed that every centrally symmetric convex progression C⊂ℤ^d is contained in a generalised arithmetic progression of size d^O(d2) # C. Berg and Henk improved the size bound to d^{O(dlog 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.

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