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### A Cantor set in the plane that is not σ-monotone

Fundamenta Mathematicae

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

### A continuity property of the dimension of the harmonic measure of Cantor sets under perturbations

Annales de l'I.H.P. Probabilités et statistiques

### A criterion for pure unrectifiability of sets (via universal vector bundle)

Annales Polonici Mathematici

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi }_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^{m}$-measurable subset of ℝⁿ with ${}^{m}\left(A\right)<\infty$. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $\left(V,v\right)|V\in G\left(n,m\right),v\in V$ such that, for all P ∈ A, one has ${}^{m\left(n-m\right)}\left(V\in G\left(n,m\right)|\left(V,{\pi }_{V}\left(P\right)\right)\in Z\right)>0$. One can replace “for all P ∈ A” by “for ${}^{m}$-a.e. P ∈...

### A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic

Open Mathematics

We study a parameterized family of singular functions which appears in a paper by H. Okamoto and M. Wunsch (2007). Various properties are revisited from the viewpoint of fractal geometry and probabilistic techniques. Hausdorff dimensions are calculated for several sets related to these functions, and new properties close to fractal analysis and strong negations are explored.

### A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces

Studia Mathematica

Let Γ be a closed set in ${ℝ}^{n}$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants ${c}_{1}>0$ and ${c}_{2}>0$ such that ${c}_{1}{r}^{d}\le µ\left(B\left(x,r\right)\right)\le {c}_{2}{r}^{d}$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces ${L}_{p}\left(\Gamma \right)$, 0 < p ≤ ∞, with respect to...

### A generalized $\sigma$-porous set with a small complement.

Abstract and Applied Analysis

### A method for evaluating the fractal dimension in the plane, using coverings with crosses

Fundamenta Mathematicae

Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set ${E}_{\epsilon }$, we consider a new method based upon coverings of ${E}_{\epsilon }$ with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals,...

### A multifractal analysis of an interesting class of measures

Colloquium Mathematicae

### A relation between dimension of the harmonic measure, entropy and drift for a random walk on a hyperbolic space.

Electronic Communications in Probability [electronic only]

### A Sard type theorem for Borel mappings

Colloquium Mathematicae

We find a condition for a Borel mapping $f:{ℝ}^{m}\to {ℝ}^{n}$ which implies that the Hausdorff dimension of ${f}^{-1}\left(y\right)$ is less than or equal to m-n for almost all $y\in {ℝ}^{n}$.

### A Short Proof of a Theorem of Ruelle.

Monatshefte für Mathematik

### A uniform dimension result for two-dimensional fractional multiplicative processes

Annales de l'I.H.P. Probabilités et statistiques

Given a two-dimensional fractional multiplicative process ${\left({F}_{t}\right)}_{t\in \left[0,1\right]}$ determined by two Hurst exponents ${H}_{1}$ and ${H}_{2}$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $\left[0,1\right]$ by $F$ if and only if ${H}_{1}={H}_{2}$.

### A Variational Principle for the Hausdorff Dimension of Fractal Sets.

Mathematica Scandinavica

### About the $h$-measure of a set. II.

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

### Absolut konvergente Reihen und das Hausdorffsche Mass

Czechoslovak Mathematical Journal

### AC-устранимость, хаусдорфова размерность и $\left(N\right)$ свойство.

Sibirskij matematiceskij zurnal

### Additive Gruppen mit vorgegebener Hausdorffscher Dimension.

Journal für die reine und angewandte Mathematik

### An analytic study on the self-similar fractals: Differentiation of integrals.

Collectanea Mathematica

### An elementary proof of Hawkes's conjecture on Galton-Watson trees.

Electronic Communications in Probability [electronic only]

### An improved bound on the Minkowski dimension of Besicovitch sets in ${ℝ}^{3}$.

Annals of Mathematics. Second Series

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