A d-dimensional analyst's Travelling Salesman theorem for arbitrary sets in Euclidean space

In this thesis, we discuss recent progress on higher dimensional analogues to the Analyst’s Travelling Salesman Theorem (TST) of Peter Jones. The TST characterizes subsets of rectifiable curves in the plane, via a multiscale sum of β-numbers. These β-numbers measure how far a set E deviates from a straight line at a particular scale and location. This idea was extended by Okikiolu to subsets of Euclidean space and by Schul to subsets of Hilbert space. In 2018, Azzam and Schul introduced a variant of the Jones β-number. With this, they, and separately Villa, proved similar results for higher dimensional subsets of Euclidean space. In particular, Villa characterizes the lower regular subsets of a certain class of d-dimensional surfaces, first introduced in a 2004 paper of David. We prove an analogous result for arbitrary subsets of Euclidean space, we do not need to assume lower regularity. To do this, we introduce a new d-dimensional variant of the Jones β-number that is defined for any set in Euclidean space. A significant portion of this thesis is dedicated to studying the properties of this β-number.2022-11-2