Discrete Schrödinger operators with decaying and oscillating potentials

We study a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential $V(n)=\lambda n^{-\alpha}\cos(\pi \omega n^\beta)$, with $1<\beta<2\alpha$, we prove that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.