  We exhibit subshifts admitting weakly mixing (probability) measures, for
arbitrary $\epsilon > 0$, with word complexity $p$ satisfying $\limsup
\frac{p(q)}{q} < 1.5 + \epsilon$. For arbitrary $f(q) \to \infty$, said
subshifts can be made to satisfy $p(q) < q + f(q)$ infinitely often.
  We establish that every subshift associated to a rank-one transformation (on
a probability space) which is not an odometer satisfies $\limsup p(q) - 1.5q =
\infty$ and that this is optimal for rank-ones.
