Many point configuration questions ask how large the Hausdorff dimension of a given compact set E in some d-dimensional Euclidean space must be in order to guarantee "lots" of occurrences of some point configuration of interest. One of the most well-known examples is the Falconer distance set problem. Given a set E, its distance set is the set of distances of the form |x-y| where x and y are points in E. It is conjectured that if E has Hausdorff dimension greater than d/2, then its distance set (which is a subset of the real line) has positive Lebesgue measure.
Falconer's motivation when he originally proved positive measure for sets with dimension greater than (d+1)/2 was to generalize the Steinhaus theorem, which states that if E has positive Lebesgue measure, then the difference set contains a neighborhood of the origin. We consider a generalization along these lines. The pinned distance set at a point x is the set of distances attained when one endpoint is fixed to be x. Peres and Schlag showed that there exists a point whose pinned distance set has nonempty interior when E has dimension at least (d+2)/2. I'll discuss how to improve this to a nontrivial threshold of 7/4 in the plane. Then, I'll discuss how to extend this to more complicated point configurations, such as arbitrary trees. This is joint work with Tainara Borges, Yumeng Ou, and Eyvindur Palsson.