Intermediate Assouad-like dimensions

GARCÍA, IGNACIO; HARE, KATHRYN; MENDIVIL, FRANKLIN

Journal of Fractal Geometry

European Mathematical Society

Año: 2021

We study a class of bi-Lipschitz-invariant dimensions that range between thebox and Assouad dimensions. The quasi-Assouad dimensions and -Assouad spectrumare other special examples. These dimensions are localized, like Assouad dimensions, butvary in the depth of scale which is considered, thus they provide very refined geometricinformation. Our main focus is on the intermediate dimensions which range between thequasi-Assouad and Assouad dimensions, complementing the -Assouad spectrum whichranges 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 intervalbeing given by the quasi-Assouad and Assouad dimensions of the set. We study stabilityand continuity-like properties of the dimensions. In contrast with the Assouad-type dimensions, we see that decreasing sets in R with decreasing gaps need not have dimension 0 or1. As is the case for Hausdorff and Assouad dimensions, the Cantor set and the decreasingset have the extreme dimensions among all compact sets in R whose complementary setconsists of open intervals of the same lengths.