Related papers: Hausdorff dimension of random limsup sets
The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the…
The almost sure Hausdorff dimension of the limsup set of randomly distributed rectangles in a product of Ahlfors regular metric spaces is computed in terms of the singular value function of the rectangles.
The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions.
A formula for the Hausdorff dimension of typical limsup sets generated by randomly distributed isotropic rectangles in Heisenberg groups is derived in terms of directed singular value functions.
In this article, we prove that from any sequence of balls whose associated limsup set has full $\mu$-measure, one can extract a well-distributed subsequence of balls. From this, we deduce the optimality of various lower bounds for the…
A set of chords of a circle of given radius is represented as a metric space w.r.t. a metric introduced by Hausdorf. The form of open and closed balls with respect to this metric is established. We consider a family of Hausdorff outer…
We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekstr\"om and Persson concerning the exact value of the…
We calculate the almost sure Hausdorff dimension of uniformly random self-similar fractals. These random fractals are generated from a finite family of similarities, where the linear parts of the mappings are independent uniformly…
We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + \xi_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear…
Fine regularity of stochastic processes is usually measured in a local way by local H\"older exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian…
We study the Hausdorff dimension of Poissonian cutout sets defined via inhomogeneous intensity measures on Ahlfors-regular metric spaces. We obtain formulas for the Hausdorff dimension of such cutouts in self-similar and self-conformal…
We give a new Hausdorff content bound for limsup sets, which is related to Falconer's sets of large intersection. Falconer's sets of large intersection satisfy a content bound for all balls in a space. In comparison, our main theorem only…
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…
The mass transference principle, proved by Beresnevich and Velani in 2006, is a strong result that gives lower bounds for the Hausdorff dimension of limsup sets of balls. We present a version for limsup sets of open sets of arbitrary shape.
This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family $P_{\lambda}:\mathbb{R}^{n}\to\mathbb{R}^{m}, \lambda \in \Lambda,$ of orthogonal…
Let $\{X_n= e^{2\pi i \theta_n}\}$ be a sequence of Steinhaus random variables, where $\theta_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random…
Let $\{x\_n\}\_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda\_n\} \_{n\geq 0}$ a sequence of positive real numbers converging to 0, and $\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$ and $\alpha>0$.…
We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given $d\in [0,2]$ we show that there exists such a meromorphic…
We consider the Ekst\''om-Persson conjecture concerning the value of the Hausdorff dimension of random covering sets formed by balls with radii $(k^{-\alpha})_{k=1}^\infty$ and centres chosen independently at random according to an…
In this paper, we investigate the Hausdorff dimension of naturally occurring sets of inhomogeneous well-approximable points with a sequence of real invertible matrices $\mathcal{A}=(A_n)_{n\in\mathbb{N}}$. Specifically, for a given point…