Related papers: Heterogeneous ubiquitous systems in $\mathbb{R}^{d…

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…

For integer $m\ge3$, we study the dynamical system $(\Lambda_m,\sigma_m)$ where $\Lambda_m$ is the set $\{w\in\{0,1\}^\mathbb{N}: w$ does not contain $0^m$ or $1^m\}$ and $\sigma_m$ is the shift map on $\{0,1\}^\mathbb{N}$ restricted to…

Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable'…

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…

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 $\nu_\lambda^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known…

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $\beta>1$ let $T_{\beta}$ be the $\beta$-transformation on $[0,1]$. We determine the Lebesgue measure and Hausdorff…

We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

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…

Given $\rho\in(0, 1/3]$, let $\mu$ be the Cantor measure satisfying $\mu=\frac{1}{2}\mu f_0^{-1}+\frac{1}{2}\mu f_1^{-1}$, where $f_i(x)=\rho x+i(1-\rho)$ for $i=0, 1$. The support of $\mu$ is a Cantor set $C$ generated by the iterated…

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable…

Let Q be an infinite set of positive integers. Denote by W_{\tau, n}(Q) (resp. W_{\tau, n}) the set of points in dimension n simultaneously \tau--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial…

In this paper, we study the uniform random covering problem in general metric space $(X,d)$. Let $\omega=(\omega_n)_{n\in\mathbb N}$ be a sequence of independent identically distributed random variables on $(X,\mu)$, and…

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

Let $\mu$ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of $\mathbb R^k$. We study the Hausdorff and the packing dimensions, and the quantization dimensions…

We show that the Continuum Hypothesis implies that for every $0<d_1\leq d_2<n$ the measure spaces $(\RR^n,\iM_{\iH^{d_1}},\iH^{d_1})$ and $(\RR^n,\iM_{\iH^{d_2}},\iH^{d_2})$ are isomorphic, where $\iH^d$ is $d$-dimensional Hausdorff measure…

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $\pi^{{\bf a}}:\; \Sigma=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the…

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…

We investigate the size and large intersection properties of $$E_{t}=\{x\in\R^d \:|\: \|x-k-x_{i}\|<{r_{i}}^t\text{for infinitely many}(i,k)\in I^{\mu,\alpha}\times\Z^d\},$$ where $d\in\N$, $t\geq 1$, $I$ is a denumerable set,…