  For strongly screened Coulomb interactions, quantum Hall interferometers can
operate in a novel regime: the intrinsic energy gap can be larger than the
charging energy, and addition of flux quanta can occur without adding
quasi-particles. We show that flux superperiods are possible, and reconcile
their appearance with the Byers-Yang theorem. We explain that the observation
of anyonic statistical phases is possible by tuning to the transition from a
regime with constant chemical potential to a regime with constant particle
density, where a flux superperiod changes to a periodicity with one flux
quantum at a critical magnetic field strength.
