In a quasi-crystal, there is no translational invariance. This means there is no delta-function in the Fourier transform. But to get a sharp scattering spot, we need a delta function.
Physically, in a crystal, we get sharp scattering spots because it is possible to get all the scattering waves from each cell in phase. But for a quasi-crystal, apparently, this is impossible.
A quasi-crystal has no long-range translational symmetry, but it does have long range orientational symmetry. The lattice sites all occur at well defined angles, and in well-defined planes. It is reflection from these planes that causes the well-defined spots. The fact that there is no translational symmetry within those planes does not bear on the interference phenomena perpendicular to the plane. A collimated ("spot") input beam will be reflected into a collimated "spot" output beam.
