  Marstrand's celebrated projection theorem gives the Hausdorff dimension of
the orthogonal projection of a Borel set in Euclidean space for almost all
orthogonal projections. It is straightforward to see that sets for which the
Fourier and Hausdorff dimension coincide have no exceptional projections, that
is, \emph{all} orthogonal projections satisfy the conclusion of Marstrand's
theorem. With this in mind, one might believe that the Fourier dimension (or at
least, Fourier decay) could be used to give better estimates for the Hausdorff
dimension of the exceptional set in general. We obtain projection theorems and
exceptional set estimates based on the Fourier spectrum; a family of dimensions
that interpolates between the Fourier and Hausdorff dimensions. We apply these
results to show that the Fourier spectrum can be used to improve several
results for the Hausdorff dimension in certain cases, such as Ren--Wang's sharp
bound for the exceptional set in the plane, Peres--Schlag's exceptional set
bound and Bourgain--Oberlin's sharp $0$-dimensional exceptional set estimate.
