What determines the spreading of a wave packet?

The multifractal dimensions D-2(mu) and D-2(psi) of the energy spectrum and eigenfunctions, respectively, are shown to determine the asymptotic scaling of the width of a spreading wave packet. For systems where the shape of the wave packet is preserved, the kth moment increases as t(k beta) with beta = D-2(mu) / D-2(psi), while, in general, t(k beta) is an optimal lower bound. Furthermore, we show that in d dimensions asymptotically in time the center of any wave packet decreases spatially as a power law with exponent D-2(psi) - d, and present numerical support for these results.