Sharp bounds for the Tao-Vu Discrete John's Theorem
- Resource Type
- Working Paper
- Authors
- van Hintum, Peter; Keevash, Peter
- Source
- Subject
- Mathematics - Combinatorics
05B35, 05B40, 11P21, 11P70, 52A27, 52A40, 52C07, 52C17
- Language
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.
Comment: 2 pages