Discrete & Computational Geometry

, Volume 58, Issue 3, pp 596–613 | Cite as

A Minkowski Theorem for Quasicrystals

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Abstract

The aim of this paper is to generalize Minkowski’s theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in \(\mathbf {R}^n\). In some situations, one may replace the lattice by a more general set for which a notion of density exists. In this paper, we prove a Minkowski theorem for quasicrystals, which bounds from below the frequency of differences appearing in the quasicrystal and belonging to a centrally symmetric convex body. The last part of the paper is devoted to quite natural applications of this theorem to Diophantine approximation and to discretization of linear maps.

Keywords

Minkowski theorem Quasicrystals Diophantine approximation Discretization 

Mathematics Subject Classification

05A20 11B05 52C23 11H06 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade Federal FluminenseNiteroiBrazil
  2. 2.IMJ-PRG, 4 place Jussieu, case 247Paris Cedex 05France
  3. 3.Modal’X, Bureau E08, Bâtiment GUniversité Paris OuestNanterreFrance

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