Generalised lower Assouad-type dimensions and their interpolations

TL;DR

This paper introduces and analyzes the $\,\phi$-lower Assouad dimension, a generalized fractal dimension, establishing its properties, equivalences, and interpolation behaviors to deepen understanding of fractal geometry.

Contribution

It extends the lower Assouad dimension to a generalized form, proving key properties, equivalences, and interpolation results that enhance the theoretical framework of fractal dimensions.

Findings

Established equivalence of $\,\phi$-lower Assouad dimensions with dimension functions

Proved analytic properties related to regularity of the $\,\phi$-lower dimension

Analyzed positive and negative interpolation properties of the $\,\phi$-lower dimension

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.