Persistence of gaps in the interacting Hofstadter model

The energy spectrum of the Hofstadter model has a fractal structure with infinitely many gaps. We prove the persistence of each gap in presence of Hubbard interaction in the case of small transversal hopping, even when the coupling is much larger than the non interacting gaps. The proof relies on a subtle interplay of Renormalization Group arguments combined with number-theoretic properties of the incommensurate frequencies.