Finding Patterns and Arithmetic Progressions in Fractal Sets

A vibrant and classic area of research is that of relating the size of a set to the finite point configurations that it contains.  This includes chains, trees, graphs with loops, as well as arithmetic progressions. In the fractal setting, size may refer to dimension or measure.  In this talk, we will consider two notions of size- Hausdorff dimension and Newhouse thickness- that can be used to guarantee the existence of arbitrarily long paths within fractal subsets of Euclidean space.  We also consider some preliminary results for arithmetic progressions occurring within fractal sets.  (This talk is based on joint works with Alex Mcdonald and Samantha Sandberg).