Interpolating with generalized Assouad dimensions

The $\varphi$ -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 $\varphi$ -Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space F and $\alpha \in R$ satisfying ${\overline{\text{dim}}}_{\text{B}}F<\alpha \le {\text{dim}}_{\text{A}}F$ that there is a function $\varphi$ so that the $\varphi$ -Assouad dimension of F is equal to $\alpha$ . We further show that the "upper" variant of the dimension is fully determined by the $\varphi$ -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 $\varphi$ -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 $\varphi$ -Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.