  We study the rotational properties of a two-component Bose-Einstein condensed
gas of distinguishable atoms which are confined in a ring potential using both
the mean-field approximation, as well as the method of diagonalization of the
many-body Hamiltonian. We demonstrate that the angular momentum may be given to
the system either via single-particle, or "collective" excitation. Furthermore,
despite the complexity of this problem, under rather typical conditions the
dispersion relation takes a remarkably simple and regular form. Finally, we
argue that under certain conditions the dispersion relation is determined via
collective excitation. The corresponding many-body state, which, in addition to
the interaction energy minimizes also the kinetic energy, is dictated by
elementary number theory.
