  We give the best possible upper bound for the number of exceptional values of
the Lagrangian Gauss map of complete improper affine fronts in the affine
three-space. We also obtain the sharp estimate for weakly complete case. As an
application of this result, we provide a new and simple proof of the parametric
affine Bernstein problem for improper affine spheres. Moreover we get the same
estimate for the ratio of canonical forms of weakly complete flat fronts in
hyperbolic three-space.
