  We obtain (up to logarithmic scaling) the power-law lower bound
$M_{p}(T_{k})\gtrsim T_{k}^{(1-\delta)p}$ on a subsequence
$T_{k}\rightarrow\infty$, uniformly across $p>0$, for discrete one-dimensional
quasiperiodic Schr\"odinger operators with frequencies satisfying
$\beta(\alpha)>\frac{3}{\delta}\min_{\sigma}\gamma$. We achieve this by
obtaining a quantitative ballistic lower bound for the Abel-averaged time
evolution of general periodic Schr\"odinger operators in terms of the
bandwidths. A similar result without uniformity, which assumes
$\beta(\alpha)>\frac{C}{\delta}\min_{\sigma}\gamma$, was obtained earlier by
Jitomirskaya and Zhang, for an implicit constant $C<\infty$.
