We give an entirely new approach to the problem of mutually unbiased bases
(MUBs), based on a Fourier analytic technique in additive combinatorics. The
method provides a short and elegant generalization of the fact that there are
at most d+1 MUBs in \Co^d. It may also yield a proof that no complete
system of MUBs exists in some composite dimensions -- a long standing open
problem.Comment: 11 page