Problems related to uniform rectifiability and bi-Lipschitz images

Abstract

In the first part of this thesis we give a new sufficient condition for a set to be uniformly d-rectifiable. The condition is that through each point of the set, there are d-many uniformly spread out line segments contained in the set. This is a simpler version of the conjecture posed by Azzam in 2021, where he asked whether the so-called Many Segments property implies uniform rectifiability. The second part of the thesis is concerned about the bounds on the dimension of the intersection of images of L-bi-Lipschitz maps with c-lower content s-regular subsets of a Euclidean space. The result of Mattila and Saaranen gives us a lower bound in terms of L,c,s and the dimensions whereas the upper bound is obtained by an explicit calculation in the case of an s-dimensional Cantor set. We also give a sufficient geometric condition for a set E in Rn, which guarantees that the image of an L-bi-Lipschitz map from Rd into Rn intersects E in a set of dimension strictly less than d.