Generalized fractal measures on Cartesian products: Critical cases and multidimensional Cantor sets

This paper investigated the interplay of generalized fractal measures, specifically packing and Hewitt-Stromberg measures, on Cartesian products of symmetric generalized Cantor sets. By developing a unified framework for analyzing product sets in separable metric spaces, we uncovered how these measures interact in both typical and critical scenarios. Through explicit construction of multidimensional Cantor sets, we demonstrated instances where classical inequalities break down, particularly in cases involving vanishing and divergent measures. Our results enhance the theoretical foundation for multifractal analysis, offering new tools to address complexity in product geometries.