Interpolating with generalized Assouad dimensions

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 ${\stackrel{}{\dim }}_{\mathrm{B}}F<\alpha \le {\dim }_{\mathrm{A}}F$ 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 Galton–Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. This result follows from two results which may be of general interest: a sharp large deviations theorem for Galton–Watson processes with bounded offspring distribution, and a 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.