Besicovitch Sets, Distance Sets, and Embeddings into Euclidean Spaces Without Shrinking

In this thesis, we study problems related to scaled oscillation, Besicovitch sets, Falconer distance problem and Sard's theorem. In chapter 2 (joint work with Marianna Csörnyei and Bobby Wilson), we study the size and regularity properties of level sets of continuous functions with bounded lower-scaled oscillation. In chapter 3 (joint work with Marianna Csörnyei and Kornélia Héra), we consider the Hausdorff dimension of planar Besicovitch-type sets for rectifiable sets. In chapter 4, we construct a non-trivial 1-rectifiable set in the plane, for which there exists a 1-dimensional Besicovitch set. In chapter 5 (joint work with Ryan Bushling and Bobby Wilson), we use techniques of algorithmic complexity theory to study distance sets. In chapter 6 (joint work with Ryan Bushling and Bobby Wilson), we study the packing dimension of distance sets with respect to polyhedral norms. In chapter 7 (joint work with Marianna Csörnyei) we study which spaces can be embedded into Euclidean space without decreasing any of the distances. This work is motivated by the following generalization of the classical Sard theorem in the plane.