Assouad-type dimensions and their role in embedding metric spaces 

Abstract: The Assouad dimension is a measure of maximal local `thickness' of metric spaces and plays a role in studying bi-Lipschitz embeddings. We will motivate its definition, introduce several variants and apply them to the study of sets generated by stochastic processes. Chiefly, we will look at Mandelbrot percolation and general fractal percolation which can be modelled by an appropriate Galton-Watson process. For Mandelbrot percolation in R^d, the obtained set can be made arbitrarily small `small’ in the sense of most classical dimensions such as the Hausdorff dimension, yet we show that it can almost surely not be embedded into \$R^k\$ for \$k < n\$.