Intermediate Assouad-like dimensions

Abstract

We study a class of bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-Assouad spectrum are other special examples. These dimensions are localized, like Assouad dimensions, but vary in the depth of scale which is considered, thus they provide very refined geometric information. Our main focus is on the intermediate dimensions which range between the quasi-Assouad and Assouad dimensions, complementing the $\theta$-Assouad spectrum which ranges between the box and quasi-Assouad dimensions. 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 stability and 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. 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.