Quasicrystals and almost periodicity
type: Dataset
description: We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let U denote the space of uniformly discrete subsets of the Euclidean space. Let A denote the elements of U that admit an autocorrelation measure. A Patterson set is an element of A such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that S in A is a Patterson set if and only if S is almost periodic in (U,T), where T denotes the Besicovitch topology. Let chi be an ergodic random element of U. We prove that chi is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum. As an illustration, we study deformed model sets.
landing page: 0010-3616
keyword: 52c23-60g55
keyword: almost-periodicity
keyword: dynamical-system
keyword: harmonic-analysis
keyword: infoeu-reposemanticsarticle
keyword: journal-articles
keyword: mathmath-mpmathematics-mathmathematical-physics-math-ph
keyword: mathmath-prmathematics-mathprobability-mathpr
keyword: palm-measure
keyword: quasicrystal
keyword: spectral-measure
keyword: stochastic-geometry
issued: 2026-05-14T14:38:42.335559
modified: 2026-05-14T14:38:42.335565
language: en
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

Quasicrystals and almost periodicity
type: Distribution
format: HTML
access URL: https://hal.science/hal-00078642
issued: 2026-05-14T14:38:42.339891
modified: 2026-05-14T14:38:42.305677