The Assouad dimension is a notion of dimension in fractal geometry that has been the
subject of much interest in recent years. This book, written by a world expert on the topic, is …

For the spherical mean operators $\scr {A} _t $ in $\Bbb {R}^ d $, $ d\ge 2$, we consider the
maximal functions $ M_Ef=\sup_ {t\in E}|\scr {A} _t f| $, with dilation sets $ E\subset [1, 2] $. In …

Dimension theory lies at the heart of fractal geometry and concerns the rigorous
quantification of how large a subset of a metric space is. There are many notions of …

We investigate the distortion of Assouad dimension and the Assouad spectrum under
Euclidean quasiconformal maps. Our results complement existing conclusions for Hausdorff …

The intermediate dimensions are a family of dimensions which interpolate between the
Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a …

We introduce a family of dimensions, which we call the Φ-intermediate dimensions, that lie
between the Hausdorff and box dimensions and generalise the intermediate dimensions …

Let $ d\in\mathbb {N} $ and let $\varphi\colon (0, 1)\to [0, d] $. We prove that there exists a
set $ F\subset\mathbb {R}^ d $ such that $\operatorname {dim} _A^\theta F=\varphi (\theta) …

We study the dimension theory of limit sets of iterated function systems consisting of a
countably infinite number of conformal contractions. Our focus is on the Assouad type …

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad
dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad …

The $\phi $-Assouad dimensions are a family of dimensions which interpolate between the
upper box and Assouad dimensions. They are a generalization of the well-studied Assouad …