Green's function estimates for quasi-periodic operators on ${\mathbb{Z}}^d$ with power-law long-range hopping

We establish quantitative Green’s function estimates for a class of quasi-periodic (QP) operators on ${\mathbb{Z}}^d$ with power-law long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic version of localization, the finite volume version of $(\frac{1}{2}-)$-Hölder continuity of the IDS, and the absence of eigenvalues (for Aubry dual operators).