Intermediate Assouad-like dimensions

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