A Fourier analytic approach to the problem of mutually unbiased bases

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 ${\text{Co}}^d$. It may also yield a proof that no complete system of MUBs exists in some composite dimensions – a long standing open problem.