Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation

In this paper, we study the multi-frequency quasi-periodic operator with a Gevrey type perturbation. We first establish the large deviation theorem (LDT) for the multi-dimensional operator with a sub-exponential (or Gevrey) long-range hopping and then prove the pure point spectrum property. Based on the LDT and the Aubry duality, we show the absence of point spectrum for the $1D$ exponential long-range operator with a multi-frequency and a Gevrey potential. We also prove the spectrum has the positive Lebesgue measure.