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Sharp bounds for a discrete John’s theorem
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 4 / July 2024
- Published online by Cambridge University Press:
- 05 March 2024, pp. 484-486
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- Article
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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.
