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We study measures on the real line and present various versions of what itmeans for such a measure to take only finitely many values. We then studyperturbations of the Laplacian by such measures. Using Kotani-Remling theory,we show that the resulting operators have empty absolutely continuous spectrumif the measures are not periodic. When combined with Gordon type arguments thisallows us to prove purely singular continuous spectrum for some continuummodels of quasicrystals.

Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals