Interpolating with generalized Assouad dimensions

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

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