Central limit theorem for generalized Weierstrass functions

Let $f$ be a $C^{2+epsilon}$ expanding map of the circle and $v$ be a $C^{1+epsilon}$ real function of the circle. Consider the twisted cohomological equation $v(x) = alpha (f(x)) - Df(x) alpha (x)$ which has a unique bounded solution $alpha$. We prove that $alpha$ is either $C^{1+epsilon}$ or nowhere differentiable, and if $alpha$ is nowhere differentiable then the Newton quotients of $alpha$, after an appropriated normalization, converges in distribution to the normal distribution, with respect to the unique absolutely continuous invariant probability of $f$.