Complexity results for CR mappings between spheres

Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation. We then sharpen our results and obtain various congruences guaranteeing nonuniqueness. We also show that a gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem.