Persistence of gaps in the interacting anisotropic Hofstadter model

We consider an interacting version of the Hofstadter model, which in the absence of interactions has a spectrum given by a Cantor set, provided that the adimensional parameter α is an irrational number. In the anisotropic situation where the hopping t2 is smaller then t1, we rigorously prove that the nth gap persists in the presence of interaction, even for interactions much stronger than the gap. We assume a Diophantine property for α and that t2/t1,U/t1 are positive and smaller than some constant, weakly depending on n. The proof relies on a subtle interplay of renormalization group arguments combined with number-theoretic properties.