## Moscow Journal of Combinatorics and Number Theory

### Discrete analogues of John's theorem

#### Abstract

As a discrete counterpart to the classical theorem of Fritz John on the approximation of symmetric $n$-dimensional convex bodies $K$ by ellipsoids, Tao and Vu introduced so called generalized arithmetic progressions $P(A,b)⊂ℤn$ in order to cover (many of) the lattice points inside a convex body by a simple geometric structure. Among others, they proved that there exists a generalized arithmetic progressions $P(A,b)$ such that $P(A,b)⊂K∩ℤn⊂ P(A,O(n)3n∕2b)$. Here we show that this bound can be lowered to $nO(lnn)$ and study some general properties of so called unimodular generalized arithmetic progressions.

#### Article information

Source
Mosc. J. Comb. Number Theory, Volume 8, Number 4 (2019), 367-378.

Dates
Revised: 27 May 2019
Accepted: 16 June 2019
First available in Project Euclid: 29 October 2019

https://projecteuclid.org/euclid.moscow/1572314453

Digital Object Identifier
doi:10.2140/moscow.2019.8.367

Mathematical Reviews number (MathSciNet)
MR4026544

Zentralblatt MATH identifier
07126249

#### Citation

Berg, Sören Lennart; Henk, Martin. Discrete analogues of John's theorem. Mosc. J. Comb. Number Theory 8 (2019), no. 4, 367--378. doi:10.2140/moscow.2019.8.367. https://projecteuclid.org/euclid.moscow/1572314453

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