UBC Math Department Colloquium: Krystal Taylor

A vibrant and classic area of research is that of relating the size of a set to the finite point configurations that it contains. Here, size may refer to cardinality, dimension, or measure. It is a consequence of the Lebesgue density theorem, for instance, that sets of positive measure in ${\mathbb{R}}^n$contain a similar copy (and all sufficiently small scalings) of any given finite point configuration. In another direction, a seminal result of Szemerédi demonstrates the existence of arithmetic progressions in subsets of $\mathbb{N}$ with positive upper density. In the fractal setting, there is a rich literature on finite point configurations, and some notions of size are more appropriate for certain settings. For example, it is known that full Hausdorff dimension is not enough to guarantee the existence of a 3-term arithmetic progression in subsets of ${\mathbb{R}}^n$. Even full Hausdorff dimension and maximal Fourier dimension are not enough, as demonstrated by Shmerkin in the line, following up on a related result of Laba and Pramanik. However, sufficient thickness suffices, as demonstrated by Yavicoli. In this talk, we give an introduction to some notions of size and dimension that are robust in the fractal setting. In particular, we consider how Hausdorff dimension and Newhouse thickness can be used to guarantee the existence and abundance of arbitrarily long paths within fractal subsets of Euclidean space.