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**Interpolating with generalized Assouad dimensions.**
arXiv:2308.12975

Preprint, arXiv:2308.12975 [math.CA] (2023).

Summary: 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 \(\overline{\operatorname{dim}}_{\mathrm{B}}F<\alpha\leq\operatorname{dim}_{\mathrm{A}} F\) 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.

### MSC:

28A80 | Fractals |

37C45 | Dimension theory of smooth dynamical systems |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

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