According to a classical theorem of Steinhaus, if a subset E of d-dimensional Euclidean space

has positive Lebesgue measure, then its difference set, E-E, contains a neighbourhood of the

origin. Configuration set problems concern characterizing the sizes of collections of point

configurations that arise among the points of a subset of Euclidean (or more general) spaces; there are versions in discrete geometry (Erd\"os-Purdy type problems) for sets with a large number N of points, as well as in continuous geometry (Falconer type problems) for fractals with some lower bound on their dimension. An example of the latter is a result of Mattila and Sj\"olin that if the Hausdorff dimension of E is >(d+1)/2, then the set of distances between points of E contains an open interval. I will discuss a general approach to results of this type using Fourier integral operators, giving positive results for a wide variety of geometries, and also discuss examples where it fails.

This is joint work with Alex Iosevich and Krystal Taylor.