Generalised lower Assouad-type dimensions and their interpolations

Abstract

This paper investigates the analytic and structural properties of the $\phi$-lower Assouad dimension, a generalized notion extending the lower Assouad dimension. We establish the equivalence of $\phi$-lower Assouad dimensions with respect to the dimension functions, prove analytic properties related to the regularity of the $\phi$-lower dimension, and analyse the role of rate windows in this context. Furthermore, we explore both positive and negative interpolation properties of the $\phi$-lower dimension by presenting corresponding theorems that delineate these behaviors.