  We use exponential sums to study the fractal dimension of the graphs of
solutions to linear dispersive PDE. Our techniques apply to Schr\"odinger,
Airy, Boussinesq, the fractional Schr\"odinger, and the gravity and
gravity-capillary water wave equations. We also discuss applications to certain
nonlinear dispersive equations. In particular, we obtain bounds for the
dimension of the graph of the solution to cubic nonlinear Schr\"odinger and
Korteweg-de Vries equations along oblique lines in space-time.
