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 spectrum with a more general form of scale sensitivity that is often closely related to “phase-transition” phenomena in sets.
In this article we establish a number of key properties of the \(\phi \) -Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space F and \(\alpha \in {\mathbb {R}}\) satisfying \(\overline{\textrm{dim}}_{\textrm{B}}F<\alpha \le \textrm{dim}_{\textrm{A}}F\) that there is a function \(\phi \) so that the \(\phi \) -Assouad dimension of F is equal to \(\alpha \) . We further show that the “upper” variant of the dimension is fully determined by the \(\phi \) -Assouad dimension, and that homogeneous Moran sets are in a certain sense generic for these dimensions.
Further, we study explicit examples of sets where the Assouad spectrum does not reach the Assouad dimension. We prove a precise formula for the \(\phi \) -Assouad dimensions for the boundary of Galton–Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. The proof of this result combines a sharp large deviations theorem for Galton–Watson processes with bounded offspring distribution and a general Borel–Cantelli-type lemma for infinite structures in random trees. Finally, we obtain results on the \(\phi \) -Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.
