Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine aprroximation properties