On the study of general multifractal analysis of sets using vector-valued functions

In this work, we present a general approach to computing the Hewitt-Stromberg dimensions of a set $ A $ in a metric space. This technique, originally introduced by Cutler and Tricot and later expanded by Ben Nasr et al., offers a generalization of classical dimension estimates and yields deeper understanding of the scaling behavior of measures in metric spaces. We applied this framework to binomial measures and developed a multifractal one for a function with respect to a gauge function $ \varphi $. Additionally, we explored the regularity of sets and measures by applying a tailored density theorem, extending existing results and offering novel insights into Moran sets.