Intermediate Assouad-like dimensions

Abstract

We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-spectrum are other special examples of these intermediate dimensions. These dimensions are localized, like Assouad dimensions, but vary in the depth of scale which is considered, thus they provide very refined geometric information. We investigate the relationship between these and the familiar dimensions. We construct a Cantor set with a non-trivial interval of dimensions, the endpoints of this interval being given by the quasi-Assouad and Assouad dimensions of the set. We study continuity-like properties of the dimensions. In contrast with the Assouad-type dimensions, we see that decreasing sets in $\mathbb{R}$ with decreasing gaps need not have dimension $0$ or $1$. Formulas are given for the dimensions of Cantor-like sets and these are used in some of our constructions. We also show that, as is the case for Hausdorff and Assouad dimensions, the Cantor set and the decreasing set have the extreme dimensions among all compact sets in $\mathbb{R}$ whose complementary set consists of open intervals of the same lengths.