Related papers: Hausdorff dimensions and Hitting probabilities for…

For given $\epsilon>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $\epsilon$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq \epsilon,$$…

We consider digits-deleted sets or Cantor-type sets with $\beta$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $\beta$. The $d$-dimentional Hausdorff measure of these sets…

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence…

By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $\beta>1$, the Hausdorff dimension of an arbitrary set in the shift space $S_\beta$ is equal to the Hausdorff dimension of its…

We study the property of hitting points for a class of $\mathbb{R}^d$-valued continuous Gaussian random fields on $\mathbb{R}^N$ with stationary increments, i.i.d. coordinates, and a regularly varying variance function $\sigma$ of index…

Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try…

Let $f: M\rightarrow M$ be a continuous map on a compact metric space $M$ equipped with a fixed metric $d$, and let $\tau$ be the topology on $M$ induced by $d$. First, we will establish some fundamental properties of the mean Hausdorff…

Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\to\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<\alpha<1$ let $I_\alpha(f)$ denote the set of…

We determine the Hausdorff and box dimension of the fractal graphs for a general class of Weierstrass-type functions of the form $f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + \theta_n)$, where $g$ is a periodic Lipschitz real function and…

Fix a space dimension $d\ge 2$, parameters $\alpha > -1$ and $\beta \ge 1$, and let $\gamma_{d,\alpha, \beta}$ be the probability measure of an isotropic random vector in $\mathbb{R}^d$ with density proportional to \begin{align*}…

We study the Macroscopic Hausdorff dimension of the upper and lower level sets of the Airy processes, following the general method developed in Khoshnevisan et al. \cite{KKX17}. For the Airy$_1$ process, the approach to macroscopic…

Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \]…

This paper studies hitting properties for the system of generalized fractional kinetic equations driven by Gaussian noise fractional in time and white or colored in space. We derive the mean square modulus of continuity and some second…

We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $\alpha$ with…

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…

Let $\theta$ be an irrational number and $\varphi: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_\varphi(\theta) =\Big\{y \in \mathbb R: \|n\theta- y\|<\varphi(n), \ {\text{for infinitely…

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in\mathbb{N},$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} {align*}…

Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in…

Hausdorff measure and Hausdorff dimension are useful tools to describe fractals. This paper investigates the bounds on the $d\log_32$-dimensional Hausdorff measure of the $d$-fold Cartesian product of the $1/3$ Cantor set, $\mathcal C^d$.…

S. Smirnov proved recently that the Hausdorff dimension of any K-quasicircle is at most 1+k^2, where k=(K-1)/(K+1). In this paper we show that if $\Gamma$ is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap \Gamma)\leq C(k) r^{1+k^2}$ for all…