  We study the groundstates of weakly interacting atomic Bose gases under
conditions of rapid rotation. We present the results of large-scale exact
diagonalisation studies on a periodic geometry (a torus) which allows studies
of compressible states with broken translational symmetry. Focusing on filling
factor $\nu=2$, we show a competition between the triangular vortex lattice, a
quantum smectic state, and the incompressible $k=4$ Read-Rezayi state. We
discuss the corrections arising from finite size effects, and the likely
behaviour for large system sizes. The Read-Rezayi state is stabilised by a
moderate amount of additional dipolar interactions.
