Jeremy Tyson

Assouad-type dimensions of graphs of functions

How do the regularity properties of a function influence the dimension of its graph? This question is classical in the case of Hausdorff or Minkowski (box-counting) dimension, and efforts to compute such dimensions for, e.g., the graphs of continuous and nowhere differentiable functions (a la Weierstrass) spurred the development of new techniques in fractal geometry and dynamical systems. In fact, the Hausdorff dimension of the graph of the Weierstrass function has only recently been determined. Assouad dimension is a quantitative and scale-invariant analog of box-counting dimension which features heavily in the modern subject of analysis in metric spaces, and the Assouad spectrum (introduced in 2018 by Fraser-Yu) is a one-parameter family of dimensions interpolating between box-counting dimension and Assouad dimension. We discuss recent work (joint with E. K. Chrontsios) providing upper bounds for the Assouad spectrum of the graphs of Holder continuous and Sobolev regular functions, along with examples demonstrating the sharpness of these bounds.