On the spectral Hausdorff dimension of 1D discrete Schr\"odinger operators under power decaying perturbations

We show that spectral Hausdorff dimensional properties of discrete Schr"oodinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component.