Mumtaz Hussain

La Trobe University · Department of Mathematics and Statistics

Given b = −A ± i with A being a positive integer, we can represent any complex number as a power series in b with coefficients in A = {0, 1,. .. , A 2 }. We prove that, for any real τ ≥ 2 and any non-empty proper subset J(b) of A, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as a power series i...

The Jarník-Besicovitch theorem is a fundamental result in metric number theory which concerns the Hausdorff dimension for certain limsup sets. We discuss the analogous problem for liminf sets. Consider an infinite sequence of positive integers, S = {q n } n∈N , exhibiting exponential growth. For a given n-tuple of functions denoted as Ψ := (ψ 1 ,....

In this paper, we investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients. For any $d\geq 1$ and $\Phi:\N\to \R_+$, our focus lies on analysing the Hausdorff dimension of the set of real numbers for which the product of mixed partial quotients $a_n(x)a_{2n}(x)\cdots a_{dn}(x)\geq...

The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable numbers and the metrical aspect of this theory leads to the study of the product of consecutive partial quotients in continued fractions. It is known that the dimension of the set of Dirichlet non-improvable numbers depends upon the number of partial quotient...

Theorems of Khintchine, Groshev, Jarn\'ik, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $\Psi$-well approximable sets. These foundational results have since been generalised to the framework of weighted Diophantine approximation for systems of real linear forms (matrices). In this article, we prov...

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximat...

We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let $A$ be matrix of real numbers, $\Psi$ an $n$-tuple of monotonic decreasing functions, and let $W_{A}(\Psi)$ be the set of points that infinitely often lie in a $\Psi(q)$-neighbourhood of the...

We develop the geometry of Hurwitz continued fractions -- a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. We obtain a detailed description of the shift space associated with Hurwitz continued fractions and, as a consequence, we contribute significantly in establishing the metrical theory...

The L\"uroth expansion of a real number $x\in (0,1]$ is the series \[ x= \frac{1}{d_1} + \frac{1}{d_1(d_1-1)d_2} + \frac{1}{d_1(d_1-1)d_2(d_2-1)d_3} + \cdots, \] with $d_j\in\mathbb{N}_{\geq 2}$ for all $j\in\mathbb{N}$. Given $m\in \mathbb{N}$, $\mathbf{t}=(t_0,\ldots, t_{m-1})\in\mathbb{R}_{>0}^{m-1}$ and any function $\Psi:\mathbb{N}\to (1,\inft...

Abstract. Let ψ : R+ → R+ be a non-increasing function. A real number x is said to be ψ-Dirichlet improvable if the system |qx − p| < ψ(t) and |q| < t has a non-trivial integer solution for all large enough t. Denote the collection of such points by D(ψ). In this paper, we prove a zero-infinity law valid for all dimension functions under natural no...

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. rein...

We prove that the set of badly approximable systems of M linear forms in N variables over the field of formal power series is hyperplane absolute winning. A consequence of this winning property is that the set has full Hausdorff dimension.

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff $f$-measure of the set of $\Psi$-approximable points on a nondegenerate manifold. We refine and extend our previous work [Int. Math. Res. Not. IMRN 2021, no. 12, 8845--8867] in which we settled the problem (for dual approximation) for hypersurfaces. We verify the GBSP for certain c...

The Jarn\'ik-Besicovitch theorem is a fundamental result in metric number theory which gives the Hausdorff dimension for limsup sets. We investigate a related problem of estimating the Hausdorff dimension of a liminf set. Let $h>0, \tau\geq 1$, and for any $j\geq 1$ define the integer sequence $q_{j+1}=q_j^h$. We prove the Hausdorff dimension of th...

Let $r=[a_1(r), a_2(r),\ldots ]$ be the continued fraction expansion of a real number $r\in \mathbb R$ . The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let $(t_1, \ldots , t_m)\in \mathbb R_+^m$ , and let $\Psi :\mathbb {N}\rightarrow (1,\infty )$ be a f...

We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(\Phi_1,\Phi_2) \defeq \EE(\Phi_1) \backslash \EE...

In this paper we consider the simultaneous approximation of real points by rational points with the error of approximation given by the functions of ‘non-standard’ heights. We prove analogues of Khintchine and Jarník-Besicovitch theorems for this setting, thus answering some questions raised by Fishman and Simmons (2017).

We consider the two-dimensional shrinking target problem in beta dynamical systems (for general $\beta>1$ ) with general errors of approximation. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in [0,1]$ , define the shrinking target set $$ \begin{align*}E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{@{}ll@{}} \lvert T_{\...

In this paper, we consider the simultaneous approximation of real points by rational points with the error of approximation given by the functions of `non-standard' heights. We prove analogues of Khintchine and Jarn\'ik-Besicovitch theorems for this setting, thus answering some questions raised by Fishman and Simmons (2017).

The classical Khintchine and Jarn\'ik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently...

Let [ a 1 ( x ), a 2 ( x ), …, a n ( x ), …] be the continued fraction expansion of an irrational number x ∈ (0, 1). The study of the growth rate of the product of consecutive partial quotients a n ( x ) a n +1 ( x ) is associated with the improvements to Dirichlet’s theorem (1842). We establish both the weak and strong laws of large numbers for th...

Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the improvements to Dirichlet's theorem (1842). We establish both the weak and strong laws of large numbers for the...

In this short note we prove a general multidimensional Jarnìk-Besicovitch theorem which gives the Hausdorff dimension of simultaneously approximable set of points with error of approximations dependent on continuous functions in all dimensions. Consequently, the Hausdorff dimension of the set varies along continuous functions. This resolves a probl...

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 enough $t$. Denote the collection of such points by $D(\psi)$. Hussain-Kleinbock-Wadleigh-Wang (2018) proved t...

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, \mu, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $\mu$-measure of the following set \[ R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law acco...

Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let $A_n(x)$ be the $n$th partial quotient of the continued fraction expansion of $x$ in the field of formal Laurent series. We cons...

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, \mu, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $\mu$-measure of the following set \[ R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law acco...

Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let A n (x) be the nth partial quotient of the continued fraction expansion of x in the field of formal Laurent series. We consider...

In this article we calculate the Hausdorff dimension of the setF(Φ)=x∈[0,1):an+1(x)an(x)⩾Φ(n)for infinitely many n∈Nandan+1(x)<Φ(n)for all sufficiently large n∈Nwhere Φ :N→(1,∞) is any function with limn→∞ Φ(n) = ∞. This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improv...

In this paper, the metric theory of Diophantine approximation associated with mixed type small linear forms is investigated. We prove Khintchine–Groshev type theorems for both the real and complex number systems. The motivation for these metrical results comes from their applications in signal processing. One such application is discussed explicitl...

In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by q↦qX+α, where q∈Zm (viewed as a row vector), X is an m×n real matrix and α∈Rn. The classical setting refers to the dist(qX+α...

We consider the two dimensional shrinking target problem in the beta dynamical system for general $\beta>1$ and with the general error of approximations. Let $f, g$ be two positive continuous functions such that $f(x)\geq g(y)$ for all $x,y\in~[0,1]$. For any $x_0,y_0\in[0,1]$, define the shrinking target set \begin{align*}E(T_\beta, f,g)=\Big\{(x,...

Let $\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ 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 $\unicode[STIX]{x1D6F9}$ -Dirichlet non-improvable numbers, $$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\u...

In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(\Phi )=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq \Phi(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< \Phi(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equati...

Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of $\psi$-approximable points on non-degenerate manifolds. The problem relates the `size' of the set of $\psi$-approximable points with the convergence or divergence of a certain series. There are...

We introduce a general principle for studying the Hausdorff measure of limsup sets. A consequence of this principle is the well-known Mass Transference Principle of Beresnevich and Velani (2006).

We introduce a general principle for studying the Hausdorff measure of limsup sets. A consequence of this principle is the well-known Mass Transference Principle of Beresnevich and Velani (2006).

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 \begin{equation*} G(\Psi ):=\Big\{x\in \lbrack 0,1):a_{n}(x)a_{n+1}(x)\,>\,\Psi \big(q_{n}(x) \big)\ \mathrm{fo...

The Generalised Baker--Schmidt Problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. There are two variants of this problem, concerning simultaneous and dual approximation. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velan...

In this paper, we investigate the two-dimensional shrinking target problem in beta-dynamical systems. Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\rightarrow \unicode[STIX]{x1D6FD}x\;\text{mod}\;1$ . Let $\unicode[STIX]{x1D6F9}_{i}$ ( $i=1,2$ ) be...

Let $\unicode[STIX]{x1D713}:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ be a non-increasing function. A real number $x$ is said to be $\unicode[STIX]{x1D713}$ -Dirichlet improvable if it admits an improvement to Dirichlet’s theorem in the following sense: the system $$\begin{eqnarray}|qx-p| has a non-trivial integer solution for all large enough $t$...

Recently, Ghosh \& Haynes \cite{HG} proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarn\'{\i}k-type result also holds for `badly approximable' points in real projective space. In particular, we prove that the natural analogue in p...

In this paper we prove an upper bound on the "size" of the set of multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower bound, providing a "zero-full" law which relates the Hausdorff measure to the convergence/divergence of a certain...

Let $\beta>1$ be a real number and define the $\beta$-transformation on $[0,1]$ by $T_\beta:x\mapsto \beta x\bmod 1$. Further, define $$W_y(T_{\beta},\Psi):=\{x\in [0, 1]:|T_\beta^nx-y|<\Psi(n) \mbox{ for infinitely many $n$}\}$$ and $$W(T_{\beta},\Psi):=\{(x, y)\in [0, 1]^2:|T_\beta^nx-y|<\Psi(n) \mbox{ for infinitely many $n$}\},$$ where $\Psi:\m...

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is also associated with a class of linear inhomogeneous partial differential equations whose solubility is related to a certain Diophantine condition. The failure of the Diophantine...

In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarn\'ik are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophanti...

In this paper we establish complete Khintchine--Groshev and Schmidt type theorems for inhomogeneous small linear forms in the so-called doubly metric case, in which the inhomogeneous parameter is not fixed.

Let W(m, n; Ψ) denote the set of Ψ1, … , Ψn–approximable points in ℝmn. The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions Ψ. Removing monotonicity from the Khintchine– Groshev theorem is attributed to different authors for different cases of m and n. It can not be removed for m = n = 1 as Duffi...

The well known theorems of Khintchine and Jarn\'ik in metric Diophantine approximation provide comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and mo...

In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine–Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximation function.

We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.

In this paper the metric theory of Diophantine approximation of linear forms that are of mixed type is investigated. Khintchine–Groshev theorems are established together with the Hausdorff measure generalizations. The latter includes the original dimension results obtained in [H. Dickinson, The Hausdorff dimension of sets arising in metric Diophant...

This paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdorff dimension results for the sets of simultaneously well-approximable points on planar curves, established in B...

In this paper we show that the set of mixed type badly approximable simultaneously small linear forms is of maximal dimension. As a consequence of this theorem we settle a conjecture of the first author.

In this paper we investigate the analogue of the classical badly approximable setup in which the distance to the nearest integer ‖⋅‖ is replaced by the sup norm |⋅|. In the case of one linear form we prove that the hybrid badly approximable set is of full Hausdorff dimension.

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 Khintchine--Groshev type theorem is established, as well as its Hausdorff measure generalization. The latter implies the co...