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We consider self-affine tilings in the Euclidean space and the associatedtiling dynamical systems, namely, the translation action on the orbit closureof the given tiling. We investigate the spectral properties of the system. Itturns out that the presence of the discrete component depends on the algebraicproperties of the eigenvalues of the expansion matrix $\phi$ for the tiling.Assuming that $\phi$ is diagonalizable over $\C$ and all its eigenvalues arealgebraic conjugates of the same multiplicity, we show that the dynamicalsystem has a relatively dense discrete spectrum if and only if it is not weaklymixing, and if and only if the spectrum of $\phi$ is a "Pisot family".Moreover, this is equivalent to the Meyer property of the associated discreteset of "control points" for the tiling.

Pisot family self-affine tilings, discrete spectrum, and the Meyer property