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

Rihab Guedri; Najmeddine Attia; Rihab Guedri; Najmeddine Attia

doi:10.3934/math.2026193

AIMS Mathematics

2026, Volume 11, Issue 2: 4739-4758. doi: 10.3934/math.2026193

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

Rihab Guedri ¹,
Najmeddine Attia ^{2
,
,}

1.
Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000 -Monastir, Tunisia
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 26 December 2025 Revised: 05 February 2026 Accepted: 09 February 2026 Published: 26 February 2026
MSC : 28A78, 28A80

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.
- generalized packing measure,
- generalized Hewitt-Stromberg measure,
- symmetric generalized Cantor sets,
- integral inequality,
- product sets
Citation: Rihab Guedri, Najmeddine Attia. Generalized fractal measures on Cartesian products: Critical cases and multidimensional Cantor sets[J]. AIMS Mathematics, 2026, 11(2): 4739-4758. doi: 10.3934/math.2026193

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Abstract

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.

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