Moscow Journal of Combinatorics and Number Theory

Discrete analogues of John's theorem

Sören Lennart Berg and Martin Henk

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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)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)Kn P(A,O(n)3n2b). 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

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

Received: 14 April 2019
Revised: 27 May 2019
Accepted: 16 June 2019
First available in Project Euclid: 29 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07] 52C07: Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21]
Secondary: 52A40: Inequalities and extremum problems

John's theorem arithmetic progressions convex bodies lattices


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.

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