On the membership of two-variable Rational Inner Functions in spaces of Dirichlet-type

We study membership of rational inner functions on the bidisk ${\mathbb{D}}^2$ in a scale of Dirichlet spaces considered by Bera, Chavan, and Ghara, and in higher-order variants of these spaces. We give a characterization for membership in terms of the geometric concept of contact order of a rational inner function at its singular points, and we further record some consequences and variants of our main result.