  We show that within any strong orbit equivalent class, there exist minimal
subshifts with arbitrarily low superlinear complexity. This is done by proving
that for any simple dimension group with unit $(G,G^+,u)$ and any sequence of
positive numbers $(p_n)_{n\in\mathbb{N}}$ such that $\lim n/p_n=0$, there exist
a minimal subshift whose dimension group is order isomorphic to $(G,G^+,u)$ and
whose complexity function grows slower than $p_n$. As a consequence, we get
that any Choquet simplex can be realized as the set of invariant measures of a
minimal Toeplitz subshift whose complexity grows slower than $p_n$.
