Related papers: The $D$-Variant of Transfinite Hausdorff Dimension

We extend the Daugavet property and a perfect version of it to transfinite cardinals in order to distinguish between spaces with the ordinary Daugavet property by some kind of complexity (topological, density\ldots), providing a number of…

Let $X$ be a ball quasi-Banach function space satisfying some mild assumptions and $H_X(\mathbb{R}^n)$ the Hardy space associated with $X$. In this article, the authors introduce both the Hardy space $H_X(\mathbb{R}^{n+1}_+)$ of harmonic…

Let $\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by…

We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski…

We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require only C^3 regularity of the maps in the IFS. The key idea,…

For any beta-shift $(X_\beta,\sigma)$ on two symbols, i.e., the symbolic coding of the beta-map for $1<\beta\leq2$, we give an exact formula for the Hausdorff dimension $\dim_{H} \Lambda_{\alpha(t)}$ as a function of $t\in\mathbb{R}$, where…

We prove that, for any Hausdorff continuum X, if dim X > 1 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then dim C(X) = 2 or C(X) is not a C-space. This generalizes results…

In this paper we prove that for a finite-dimensional real normed space $V$, every bounded mean zero function $f\in L_\infty([0,1];V)$ can be written in the form $f = g\circ T - g$ for some $g\in L_\infty([0,1];V)$ and some ergodic…

Generalizing a theorem of Ph. Dwinger, we describe the partially ordered set of all (up to equivalence) zero-dimensional locally compact Hausdorff extensions of a zero-dimensional Hausdorff space. Using this description, we find the…

Generalising a construction of Falconer, we consider classes of $G_\delta$-subsets of $\mathbb{R}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We…

In this paper we construct a new family of sets based on Diophantine approximation in the Euclidean space, and consider their applications in several problems in harmonic analysis. Our first application is on the Hausdorff dimension of our…

A breakthrough result of B\'ar\'any, Hochman and Rapaport published in 2019 established that every self-affine measure on $\mathbb{R}^2$ satisfying certain mild non-degeneracy conditions has Hausdorff dimension equal to its Lyapunov…

Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that $K$ can be identified with a subshift of finite type. With this identification, we can…

For a domain $\Omega$ in a finite-dimensional space $E$, we consider the space $M=(\Omega,d)$ where $d$ is the intrinsic distance in $\Omega$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of…

We prove that a transfinite extension of asymptotic dimension asind is trivial. We introduce a transfinite extension of asymptotic dimension asdim and give an example of metric proper space which has transfinite infinite dimension.

Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely…

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform…

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\cdots,m_1-1\} \times \{0,\cdots,m_2-1\})^{\mathbb{N}^*}$ that are invariant under the action of the semigroup of multiplicative…

We find necessary and sufficient conditions for a Lipschitz map $f:\mathbb{R}E\to X$, into a metric space to have the image with the $k$-dimensional Hausdorff measure equal zero, $H^k(f(E))=0$. An interesting feature of our approach is that…

Antoniadis, Mazur and Mottola (AMM) two years ago computed the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D Gravity. The fractal dimension was determined by the coefficient of…