Speaker
Henna Koivusalo, University of Bristol
Abstract
Badly approximable points (Bad) are the points that are the hardest to approximate with rational points or, more formally, points \(x \in \mathbb R^d\) for which for some constant \(c>0\), \(|x-p/q| > cq^{-1-1/d}, \) for all \((p/q)\in \mathbb Q^d\). They can, and have been, investigated from many perspectives, including numeration and dynamical systems as well as number theory, and they have beautiful properties no matter which way we look at them. In this talk I will discuss some of their properties that are interesting to me as a fractal geometer: What do Bad and its variants, “”look like””? What are their geometric properties, such as dimension (and beyond)? And how does Bad interact with other fractal sets? I hope to cover a few recent results, joint with subsets of Anttila, Fraser, Levesley, Ward and Zhang.