  We prove that if the frequency of the quasi-periodic $\mathrm{SL}(2,\R)$
cocycle is Diophantine, then the following properties are dense in the
subcritical regime: for any $\frac{1}{2}<\kappa<1$, the Lyapunov exponent is
exactly $\kappa$-H\"older continuous; the extended eigenstates of the potential
have optimal sub-linear growth; and the dual operator associated a subcritical
potential has power-law decay eigenfunctions. The proof is based on fibered
Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,\R)$ cocycles.
