Related papers: Dimension theory of inhomogeneous Diophantine appr…

A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a…

A classical theorem due to Mattila (see \cite{Mat84}; see also \cite{M95}, Chapter 13) says that if $A,B \subset {\Bbb R}^d$ of Hausdorff dimension $s_A, s_B$, respectively, with $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$ and $dim_{{\mathcal…

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 Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger…

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays…

We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $\epsilon$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the…

L\"uroth series, like regular continued fractions, provide an interesting identification of real numbers with infinite sequences of integers. These sequences give deep arithmetic and measure-theoretic properties of subsets of numbers…

We compute the Hausdorff dimension of the set of simultaneously $q^{-\lambda}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ for $\lambda$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given…

We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a…

We prove an analogue of a theorem of A. Pollington and S. Velani ('05), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsically approximable points on a quadratic hypersurface. The proof…

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more…

In this paper we investigate the metrical theory of Diophantine approximation associated with linear forms that are simultaneously small for infinitely many integer vectors; i.e. forms which are close to the origin. A complete…

Shrinking target problems in the context of iterated function systems have received an increasing amount of interest in the past few years. The classical shrinking target problem concerns points returning infinitely many times to a sequence…

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…

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…

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hat{\nu},$ we calculate the Hausdorff dimension of the…

Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the…

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 $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $\tau>0$, and a sequence $\{z_n\}$ of points in…

Recently Koivusalo, Levesley, Ward and Zhang introduced the set of simultaneously $\Phi$-badly approximable real vectors of $\mathbb{R}^m$ with respect to an approximation function $\Phi$, and determined its Hausdorff dimension for the…