Related papers: On Dirichlet non-improvable numbers and shrinking …

We determine the Hausdorff dimension of sets of irrationals in $(0,1)$ whose partial quotients in semi-regular continued fractions obey certain restrictions and growth conditions. This result substantially generalizes that of the second…

Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\|<Q^{-\tau}$, where $\|\cdot\|$ is the…

Let $\Psi :[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$'{th} partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$'{th} convergent. The set of $\Psi $-Dirichlet non-improvable numbers…

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: the system $$|qx-p|< \, \psi(t) \ \…

We develop the Mass Transference Principle for rectangles of Wang \& Wu (Math. Ann. 2021) to incorporate the `unbounded' setup; that is, when along some direction the lower order (at infinity) of the side lengths of the rectangles under…

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of…

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

We look at the rate of growth of the partial quotients of the infinite continued fraction expansion of an irrational number relative to the rate of approximation of the number by its convergents. In non-generic cases the Hausdorff dimension…

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $\Lambda(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in \Lambda(\mathbf{f}):…

For a norm $F$ on $\mathbb{R}^2$, we consider the set of $F$-Dirichlet improvable numbers $\mathbf{DI}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\infty$, which is a supremum norm, it is well-known that $\mathbf{DI}_F…

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the non-autonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a…

In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming…

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…

The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal…

We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued…

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for all large…

We study the continued fractions with bounded odd/even-order partial quotients. In particular, we investigate the sizes of the sets of continued fractions whose odd-order partial quotients are equal to 1. We demonstrate that the sum and the…

Let $x=[a_1(x),a_2(x),\ldots]$ be the continued fraction expansion of $x\in[0,1)$. We prove that the Hausdorff dimension of \begin{equation*}E_{even}=\{x\in[0,1)\colon a_{2n}(x)\to\infty\ (n\to\infty)\}.\end{equation*} is 1/2. In general,…

In this paper, we study inhomogeneous Diophantine approximation with rational numbers of reduced form. The central object to study is the set $W(f,\theta)$ as follows, \begin{eqnarray*} \left\{x\in [0,1]:\left…

We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…