Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2207.14654

Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets

Kathryn E. Hare·Franklin Mendivil
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we determine the almost sure values of the $Φ$-dimensions of random measures $μ$ supported on random Moran sets in $\R^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets \cite{HM} to the case of unequal scaling factors. The $Φ$-dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the $θ$-Assouad spectrum being special cases. The almost sure value of $\dim_Φμ$ exhibits a threshold phenomena, with one value for ``large'' $Φ$ (with the quasi-Assouad dimension as an example of a ``large'' dimension) and another for ``small'' $Φ$ (with the Assouad dimension as an example of a ``small'' dimension). We give many applications, including where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure $Φ$ dimension of the underlying random set is also a consequence of our results.

Explore related subjects

math.CAmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kathryn E. Hare, Franklin Mendivil. 2026-02-20. Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets. https://doi.org/10.4171/jfg%2F133

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular specular differentiation in Euclidean spaces

We study the regular specular derivative, a generalized derivative defined at every point where both one-sided derivatives exist and are finite. Geometrically, it is the slope of the mirror that reflects the left tangent ray into the right one. In one variable we derive computational formulas, prove inverse function and rotation rules, establish Quasi-Rolle's Theorem and the Quasi-Mean Value Theorem, and obtain a derivative-limit theorem, which shows that twice regularly specularly differentiable functions are continuously differentiable. We also prove both parts of the Fundamental Theorem of Calculus. In several variables we introduce specular gradients, directional derivatives, tangent hyperplanes, and normal vectors, show that a continuous specular gradient forces classical differentiability, and characterize when the specular tangent hyperplane is unique.

math.CA↗
arXiv2026

Prevalent smoothness in inhomogeneous Besov spaces

In this article, we prove that, under some assumptions on the so-called environment, prevalent functions in inhomogeneous Besov spaces recently introduced by Barral-Seuret in 2023 are multifractal, with a singularity spectrum that we determine. This completes the previous Baire generic results already obtained.

math.CA↗
arXiv2026

Lebesgue Covering Theorem and level sets of continuous functions

We formulate and prove a dimension-theoretic generalization of a version of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved algorithmically by Turzański and Ziajor, is a particular case of this result. Moreover, we study two types of sets associated with a continuous function $g \colon [0,1]^n \to \mathbb{R}$. Namely, the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ connects $i$th opposite faces of $[0,1]^n$, and the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ separates $i$th opposite faces of $[0, 1]^n$.

math.CA↗