# On a discrete John-type theorem

## TL;DR

This paper improves bounds on lattice point coverage of convex bodies using generalized arithmetic progressions, providing a discrete analogue to John’s theorem with tighter estimates and exploring properties of unimodular progressions.

## Contribution

It refines the bounds for covering lattice points in convex bodies with generalized arithmetic progressions and investigates properties of unimodular progressions.

## Key findings

- Bound on covering by progressions lowered to n^{O(ln n)}
- Established properties of unimodular generalized arithmetic progressions
- Enhanced understanding of discrete convex body approximations

## Abstract

As a discrete counterpart to the classical John theorem on the approximation of (symmetric) $n$-dimensional convex bodies $K$ by ellipsoids, Tao and Vu introduced so called generalized arithmetic progressions $P(A,b)\subset Z^n$ 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 $P(A,b)\subset K\cap Z^n\subset P(A,O(n)^{3n/2}b)$. Here we show that this bound can be lowered to $n^{O(\ln n)}$ and study some general properties of so called unimodular generalized arithmetic progressions.