Dense gaps and scaling relations in the interacting Aubry-Andre model

We study, by rigorous Renormalization Group methods, the interacting Aubry-Andre model for fermions in the extended regime. We show that the infinitely many gaps of the single particle spectrum persist in presence of weak many body interactions, despite the presence of Umklapp large momentum processes connecting the Fermi points. The width of the gaps is strongly renormalized through critical exponents which verify exact scaling relations.