  We develop a theory of interaction effects in graphene superlattices, where
tunable superlattice periodicity can be used as a knob to control the gap at
the Dirac point. Applied to graphene on hexa-boron-nitride (G/h-BN), our theory
predicts substantial many-body enhancement of this gap. Tunable by the moire
superlattice periodicity, a few orders of magnitude enhancement is reachable
under optimal conditions. The Dirac point gap enhancement can be much larger
than that of the minigaps opened by Bragg scattering at principal superlattice
harmonics. This naturally explains the conundrum of large Dirac point gaps
recently observed in G/h-BN heterostructures and their tunability by the G/h-BN
twist angle.
