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A family of non Minkowski measurable fractals in $\mathbb{R}^2$
Uta Freiberg, Jonas Lippold
https://arxiv.org/abs/2506.02640 https://…

A family of non Minkowski measurable fractals in $\mathbb{R}^2$
A long-standing conjecture of Lapidus asserts that, under certain conditions, a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in $\mathbb{R}$, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents a family of lattice-type subsets in $\mathbb{R}^2$ that are not Minkowski measurable, hence providing fu…

This https://arxiv.org/abs/2409.01292 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces
In the context of a metric measure space $(X,d,μ)$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B^θ_{p,p}(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $θ$ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical expone…

Decoding Breast Cancer in X-ray Mammograms: A Multi-Parameter Approach Using Fractals, Multifractals, and Structural Disorder Analysis
Santanu Maity, Mousa Alrubayan, Prabhakar Pradhan
https://arxiv.org/abs/2505.21080

Decoding Breast Cancer in X-ray Mammograms: A Multi-Parameter Approach Using Fractals, Multifractals, and Structural Disorder Analysis
We explored the fractal and multifractal characteristics of breast mammogram micrographs to identify quantitative biomarkers associated with breast cancer progression. In addition to conventional fractal and multifractal analyses, we employed a recently developed fractal-functional distribution method, which transforms fractal measures into Gaussian distributions for more robust statistical interpretation. Given the sparsity of mammogram intensity data, we also analyzed how variations in intens…