Arithmetic $k$-Polynomial Dynamical Localization for $k$ Power-Law Quasi-Periodic Long-Range Operators on ${\mathrm{\ell }}^2({\mathbb{Z}}^d)$

We establish a criterion of arithmetic $k$-polynomial spectral localization and $k$-polynomial dynamical localization in expectation for quasi-periodic long-range operators on ${\mathrm{\ell }}^2({\mathbb{Z}}^d)$ with power-law hopping based on the quantitative $C^k$-reducibility of the dual Schrödinger cocycle. As the application, we prove both localization properties for power-law long-range perturbations of the Almost Mathieu Operators with sufficiently large couplings and Diophantine frequencies.