Moscow Journal of Combinatorics and Number Theory

Discrete analogues of John's theorem

Sören Lennart Berg and Martin Henk

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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)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

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

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

Permanent link to this document
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

Subjects
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

Keywords
John's theorem arithmetic progressions convex bodies lattices

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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References

  • M. Alexander, M. Henk, and A. Zvavitch, “A discrete version of Koldobsky's slicing inequality”, Israel J. Math. 222:1 (2017), 261–278.
  • S. Artstein-Avidan, A. Giannopoulos, and V. D. Milman, Asymptotic geometric analysis, I, Math. Surv. Monogr. 202, Amer. Math. Soc., Providence, RI, 2015.
  • I. Bárány and A. M. Vershik, “On the number of convex lattice polytopes”, Geom. Funct. Anal. 2:4 (1992), 381–393.
  • S. L. Berg, Lattice points in convex bodies: counting and approximating, Ph.D. thesis, Technische Universität Berlin, 2018, https://tinyurl.com/bergconv.
  • U. Betke, M. Henk, and J. M. Wills, “Successive-minima-type inequalities”, Discrete Comput. Geom. 9:2 (1993), 165–175.
  • J. W. S. Cassels, An introduction to the geometry of numbers, Grundlehren der Math. Wissenschaften 99, Springer, 1959.
  • P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Math. Library 37, North-Holland, Amsterdam, 1987.
  • J. Håstad and J. C. Lagarias, “Simultaneously good bases of a lattice and its reciprocal lattice”, Math. Ann. 287:1 (1990), 163–174.
  • M. Henk, “Successive minima and lattice points”, Rend. Circ. Mat. Palermo $(2)$ Suppl. 70:1 (2002), 377–384.
  • M. A. Hernández Cifre, D. Iglesias, and J. Yepes Nicolás, “On a discrete Brunn–Minkowski type inequality”, SIAM J. Discrete Math. 32:3 (2018), 1840–1856.
  • J. C. Lagarias, H. W. Lenstra, Jr., and C.-P. Schnorr, “Korkin–Zolotarev bases and successive minima of a lattice and its reciprocal lattice”, Combinatorica 10:4 (1990), 333–348.
  • R. Malikiosis, “An optimization problem related to Minkowski's successive minima”, Discrete Comput. Geom. 43:4 (2010), 784–797.
  • R.-D. Malikiosis, “A discrete analogue for Minkowski's second theorem on successive minima”, Adv. Geom. 12:2 (2012), 365–380.
  • G. Maze, “Some inequalities related to the Seysen measure of a lattice”, Linear Algebra Appl. 433:8-10 (2010), 1659–1665.
  • D. Ryabogin, V. Yaskin, and N. Zhang, “Unique determination of convex lattice sets”, Discrete Comput. Geom. 57:3 (2017), 582–589.
  • R. Schneider, Convex bodies: the Brunn–Minkowski theory, 2nd expanded ed., Encycl. Math. Appl. 151, Cambridge Univ. Press, 2014.
  • M. Seysen, “Simultaneous reduction of a lattice basis and its reciprocal basis”, Combinatorica 13:3 (1993), 363–376.
  • T. Tao and V. Vu, Additive combinatorics, Cambridge Studies in Adv. Math. 105, Cambridge Univ. Press, 2006.
  • T. Tao and V. Vu, “John-type theorems for generalized arithmetic progressions and iterated sumsets”, Adv. Math. 219:2 (2008), 428–449.