Measures with locally finite support and spectrum

Abstract

The goal of this paper is the construction of measures μon ℝn enjoying three conflicting but fortunately compatible properties: (i) μis a sum of weighted Dirac masses on a locally finite set, (ii) the Fourier transform μˆof μis also a sumof weighted Diracmasses on a locally finite set, and (iii) μis not a generalized Dirac comb.We give surprisingly simple examples of such measures. These unexpected patterns strongly differ from quasicrystals, they provide us with unusual Poisson's formulas, and they might give us an unconventional insight into aperiodic order.