Quasicrystals and almost periodicity

Details Quasicrystals and almost periodicity Quasicrystals and almost periodicity

2002-12-03

arxiv.org/abs/math-ph/0212012v1

Physics

Mathematical Physics

Scientific paper

We introduce a topology ${\cal T}$ on the space $U$ of uniformly discrete subsets of the Euclidean space. Assume that $S$ in $U$ admits a unique autocorrelation measure. The diffraction measure of $S$ is purely atomic if and only if $S$ is almost periodic in $(U,{\cal T})$. This result relates idealized quasicrystals to almost periodicity. In the context of ergodic point processes, the autocorrelation measure is known to exist. Then, the diffraction measure is purely atomic if and only if the dynamical system has a pure point spectrum. As an illustration, we study deformed model sets.

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