  This paper aims to better understand the link better understand the links
between aperiodicity in subshifts and pattern complexity. Our main contribution
deals with substitutive subshifts, an equivalent to substitutive tilings in the
context of symbolic dynamics. For a class of substitutive subshifts, we prove a
quadratic lower bound on their pattern complexity. Together with an already
known upper bound, this shows that this class of substitutive subshifts has a
pattern complexity in $\Theta(n^2)$. We also prove that the recent bound of
Kari and Moutot, showing that any aperiodic subshift has pattern complexity at
least $mn+1$, is optimal for fixed $m$ and $n$.
