  We consider the magnetic field dependence of the chemical potential for
parabolically confined quantum dots in a strong magnetic field. Approximate
expressions based on the notion that the size of a dot is determined by a
competition between confinement and interaction energies are shown to be
consistent with exact diagonalization studies for small quantum dots. Fine
structure is present in the magnetic field dependence which cannot be explained
without a full many-body description and is associated with ground-state level
crossings as a function of confinement strength or Zeeman interaction strength.
Some of this fine structure is associated with precursors of the bulk
incompressible states responsible for the fractional quantum Hall effect.
