  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.
