  We prove a general nonlinear projection theorem for Assouad dimension. This
theorem has several applications including to distance sets, radial
projections, and sum-product phenomena. In the setting of distance sets we are
able to completely resolve the planar distance set problem for Assouad
dimension, both dealing with the awkward `critical case' and providing sharp
estimates for sets with Assouad dimension less than 1. In the higher
dimensional setting we connect the problem to the dimension of the set of
exceptions in a related (orthogonal) projection theorem. We also obtain results
on pinned distance sets and our results still hold when the distances are taken
with respect to a sufficiently curved norm. As another application we prove a
radial projection theorem for Assouad dimension with sharp estimates on the
Hausdorff dimension of the exceptional set.
