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