  We consider the Assouad dimension analogues of two important problems in
geometric measure theory. These problems are tied together by the common theme
of `passing to weak tangents'. First, we solve an analogue of Falconer's
distance set problem for Assouad dimension in the plane: if a planar set has
Assouad dimension strictly greater than 1, then its distance set has Assouad
dimension 1. We also obtain partial results in higher dimensions. Second, we
consider how Assouad dimension behaves under orthogonal projection. We extend
the planar projection theorem of Fraser and Orponen to higher dimensions,
provide estimates on the (Hausdorff) dimension of the exceptional set of
projections, and provide a recipe for obtaining results about restricted
families of projections. We provide several illustrative examples throughout.
