The Mattila-Sjölin Problem for Triangles

This dissertation contains work from the author's papers [35] and [36] with coauthor Eyvindur Palsson. The classic Mattila-Sjolin theorem shows that if a compact subset of $mathbbRd$ has Hausdorff dimension at least $frac(d+1)2$ then its set of distances has nonempty interior. In this dissertation, we present a similar result, namely that if a compact subset $E$ of $mathbbRd$, with $dgeq3$, has a large enough Hausdorff dimension then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well known Falconer distance problem which establishes a positive Lebesgue measure for the distance set instead of it having nonempty interior