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Quasicrystals and almost periodicity

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00078642
Contributor : Jean-Baptiste Gouéré <>
Submitted on : Tuesday, June 6, 2006 - 7:00:20 PM
Last modification on : Thursday, November 21, 2019 - 2:19:28 AM

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Jean-Baptiste Gouéré. Quasicrystals and almost periodicity. Communications in Mathematical Physics, Springer Verlag, 2005, 255(3), pp.655 - 681. ⟨10.1007/s00220-004-1271-8⟩. ⟨hal-00078642⟩

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