Interpolating with generalized Assouad dimensions

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

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