arXiv subjects

Mumtaz Hussain

Publications and source records attributed to Mumtaz Hussain.

At least 19 recordsLinked to original sources

arXiv2026

The Cartesian product of exact approximation sets

We determine the Hausdorff and packing dimensions of Cartesian products of one-dimensional exact approximation sets. Our main result establishes the exact-approximation counterpart of the recent product theorem of Wang and Wu (2024) for limsup approximation sets, showing that passing to the substantially smaller exact approximation sets (liminf sets) does not reduce the Hausdorff dimension of the Cartesian product. One of the key ingredients is a refinement of the well-distributed-system framework of Bandi--Ghosh--Nandi (2023) by exploiting the fine arithmetic distribution of rational points which then gives the Hausdorff dimension of the product set under a weaker convergence condition.

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arXiv2026

Well and badly approximable sets, and rapid winning

The set of $τ$-approximable numbers, $\mathcal W(τ)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad^γ$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $Ψ$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $ψ$, we prove that $\mathcal W(ψ)\cap\Bad^γ$ is strong $Ψ$-rapid winning for a natural gauge $Ψ$ determined by $ψ$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $Ψ$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $Ψ$. In particular, for $ψ(q)=q^{-τ}, τ\ge1,$ we recover the exact Jarník--Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(τ)\cap\Bad^γ\bigr)=\frac{2}{τ+1}.$$

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arXiv2026

Measure theoretic properties of large products of consecutive partial quotients

The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $φ:\mathbb{N}\to\mathbb{R}_{\geq 2}$, we determine the Lebesgue measure of the set $\mathcal{F}_{\ell}(φ)$ of irrational numbers $x$ whose regular continued fraction $x~=~[a_1(x),a_2(x),\ldots]$ is such that, for infinitely many $n\in\mathbb{N}$, there are two numbers $1\leq j<k \leq n$ satisfying \[ a_{k}(x)\cdots a_{k+\ell-1}(x) \geq φ(n), \; a_{j}(x)\cdots a_{j+\ell-1}(x) \geq φ(n). \] This result generalizes previous work by Tan and Zhou (Nonlinearity, 2024). A consequence of our result is that the strong law of large numbers for products of $\ell$ consecutive partial quotients is impossible even if the block with the largest product is removed. We also compute the Hausdorff dimension of $\mathcal{F}_3(φ)$.

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arXiv2025

Continued fractions with large prime partial quotients

We determine the Lebesgue measure and Hausdorff dimension of various sets of real numbers with infinitely many partial quotients that are both large and prime, thus extending the well-known theorems by Łuczak (1997) and Huang-Wu-Xu (2020). To this end, we obtain new asymptotics on the tail end of the almost prime zeta function. Our results include some recent work by Schindler-Zweim{ü}ller (2023).

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arXiv2025

Approximation by uniformly distributed sequences

We consider approximation properties of real points by uniformly distributed sequences. Under some assumptions on the approximation functions, we prove a Khintchine-type $0$-$1$ dichotomy law. We establish a new connection between uniform distribution and the ubiquity property. Namely, we show that a bound on the discrepancy of the sequence implies the ubiquity property, which helps to obtain divergence results. We further obtain Hausdorff dimension results for weighted sets. The key tools in proving these results are the weighted ubiquitous systems and weighted mass transference principle introduced recently by Kleinbock \& Wang, and Wang \& Wu respectively.

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arXiv2025

Weighted approximation for limsup sets

Theorems of Khintchine, Groshev, Jarník, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $Ψ$-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 prove analogues of these weighted results in a range of settings including the $p$-adics (Theorems 7 and 8), complex numbers (Theorems 9 and 10), quaternions (Theorems 11 and 12), and formal power series (Theorems 13 and 14). The key tools in proving the main parts of these results are the weighted ubiquitous systems and weighted mass transference principle introduced recently by Kleinbock--Wang [Adv. Math. (2023)] and Wang--Wu [Math. Ann. (2021)].

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arXiv2025

Metrical properties of Hurwitz Continued Fractions

We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions.%, paralleling the classical theory for regular continued fractions in real numbers. Let $Φ:\mathbb{N}\to \mathbb{R}_{>0}$ be any function. For any complex number $z$ and $n\in\mathbb{N}$, let $a_n(z)$ denote the $n$th partial quotient in the Hurwitz continued fraction of $z$. One of the main results of this paper is the computation of the Hausdorff dimension of the set \[E(Φ) := \left\{ z\in \mathbb C: |a_n(z)|\geq Φ(n) \text{ for infinitely many }n\in\mathbb{N} \right\}. \] This study is a complex analog of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5, 1319--1339].

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arXiv2025

On the Folklore set and Dirichlet spectrum for matrices

We study the Folklore set of Dirichlet improvable matrices in $\mathbb R^{m\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{ 1,1\}$ and $\{m,n\}=\{ 2,3\}$ in a constructive manner. For a wide range of integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of $0$). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Diophantine type, and then to show that most `liftings' to matrices will preserve this Diophantine type. This is a variant of a method introduced by Moshchevitin for uniform approximation. Our technique is often also applicable to arbitrary norms. As a corollary, we obtain lower bounds on the Hausdorff dimension of these sets. These statements complement previous results of the middle-named author (Selecta Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math. 2024).

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arXiv2025

A note on limsup sets of annuli

We consider the set of points in infinitely many max-norm annuli centred at rational points in $\mathbb R^{n}$. We give Jarník-Besicovitch type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension $n$, and the thickness of the annuli is decreasing rapidly then the dimension of the set tends towards $n-1$. We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's Scaling Lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021).

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arXiv2025

Transcendence and normality of complex numbers via Hurwitz continued fractions

We study the topological, dynamical, and descriptive set theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian rational. The resulting space of sequences of Gaussian integers $Ω$ is not closed. By means of an algorithm, we show that $Ω$ contains a natural subset whose closure $\overline{\mathsf{R}}$ encodes continued fraction expansions of complex numbers which are not Gaussian rationals. We prove that $(\overline{\mathsf{R}}, σ)$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

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arXiv2024

Metrical properties of finite product of partial quotients in arithmetic progressions

We investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients in continued fractions of real numbers. For any function $Φ:\mathbb N\to [2,+\infty)$ and any integer $d\geq 1$, we determine the Lebesgue measure and Hausdorff dimension of the set of real numbers for which the product of partial quotients in arithmetic progressions satisfy $a_n(x)a_{2n}(x)\cdots a_{dn}(x)\geq Φ(n)$ for infinitely many positive integers $n$. Our findings shed light on the size of the set of exceptions to Bourgain's (1988) and Host and Kra's (2005) theorems concerning the convergence of multiple ergodic averages for Gauss dynamical systems. By exploring the Hausdorff dimension of these sets, we gain valuable insights into the behaviour of such exceptions. Overall, our research contributes to a deeper understanding of the dynamics of continued fractions and their connection to the convergence properties of ergodic averages in Gauss dynamical systems.

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arXiv2023

Restricted slowly growing digits for infinite iterated function systems

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $Λ(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in Λ(\mathbf{f}): a_n(x)\in D \text{ for all }n\in\mathbb N \text{ and }\lim_{n\to\infty} a_n=\infty\}. \] For a function $φ:\mathbb{N}\to [\min D, \infty)$ such that $φ(n)\to\infty$ as $n\to\infty$, we compute the Hausdorff dimension of the set $$ S(\mathbf{f},D,φ) = \left\{ x\in \E(\mathbf{f},D) : a_n(x)\leq φ(n) \text{ for all } n\in\mathbb N \right\}. $$ We prove that the Hausdorff dimension stays the same no matter how slowly the function $φ$ grows. One of the consequences of our result is the recent work of Takahasi (2023), which only dealt with regular continued fraction expansions. We further extend our result to slowly growing products of (not necessarily consecutive) digits.

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arXiv2023

Complex numbers with a prescribed order of approximation and Zaremba's conjecture

Given $b=-A\pm i$ with $A$ being a positive integer, we can represent any complex number as a power series in $b$ with coefficients in $\mathcal A=\{0,1,\ldots, A^2\}$. We prove that, for any real $τ\geq 2$ and any non-empty proper subset $J(b)$ of $\mathcal A$, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as a power series in $b$ with coefficients in $J(b)$ and with the irrationality exponent (in terms of Gaussian integers) equal to $τ$. One of the key ingredients in our construction is the `Folding Lemma' applied to Hurwitz continued fractions. This motivates a Hurwitz continued fraction analogue of the well-known Zaremba's conjecture. We prove several results in support of this conjecture.

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arXiv2023

Liminf approximation sets for abstract rationals

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, $S=\{q_{n}\}_{n\in\mathbb{N}}$, exhibiting exponential growth. For a given $n$-tuple of functions denoted as $Ψ:=~(ψ_1, \ldots,ψ_n)$, each of the form $ψ_{i}(q)=q^{-τ_{i}}$ for $(τ_{1},\dots,τ_{n})\in\mathbb{R}^{n}_{+}$, we calculate the Hausdorff dimension of the set of points that can be $Ψ$-approximated for all sufficiently large $q\in S$. We prove this result in the generalised setting of approximation by abstract rationals as recently introduced by Koivusalo, Fraser, and Ramirez (LMS, 2023). Some of the examples of this setting include the real weighted inhomogeneous approximation, $p$-adic weighted approximation, Diophantine approximation over complex numbers, and approximation on missing digit sets.

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arXiv2023

Metrical properties of exponentially growing partial quotients

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function $Φ$, Wang-Wu theorem (2008) comprehensively describes the Hausdorff dimension of the set \begin{equation*} \EE_1(Φ):=\left\{x\in [0, 1): a_n(x)\geq Φ(n) \ {\rm for \ infinitely \ many} \ n\in \N\right\}. \end{equation*} Various generalisations of this set exist, such as substituting one partial quotient with the product of consecutive partial quotients in the aforementioned set which has connections with the improvements to Dirichlet's theorem, and many other sets of similar nature. Establishing the upper bound of the Hausdorff dimension of such sets is significantly easier than proving the lower bound. In this paper, we present a unified approach to get an optimal lower bound for many known setups, including results by Wang-Wu [Adv. Math., 2008], Huang-Wu-Xu [Israel J. Math. 2020], Bakhtawar-Bos-Hussain [Nonlinearity 2020], and several others, and also provide a new theorem derived as an application of our main result. We do this by finding an exact Hausdorff dimension of the set $$S_m(A_0,\ldots,A_{m-1}) \defeq \left\{ x\in[0,1): \, c_i A_i^n \le a_{n+i}(x) < 2c_i A_i^n,0 \le i \le m-1 \ \text{for infinitely many } n\in\N \right\},$$ where each partial quotient grows exponentially and the base is given by a parameter $A_i>1$. For proper choices of $A_i$'s, this set serves as a subset for sets under consideration, providing an optimal lower bound of Hausdorff dimension in all of them. The crux of the proof lies in introducing of multiple probability measures consistently distributed over the Cantor-type subset of $S_m(A_0,\ldots,A_{m-1})$.

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arXiv2023

Metrical properties of the product of partial quotients with geometric mean in continued fractions

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 quotients in the product string. However, one can see that the Hausdorff dimension is the same for any number of consecutive partial quotients with a constant gap. This paper is aimed at a detailed analysis on how the Hausdorff dimension changes when there is a linear gap in indices and the number of partial quotients in the product grows. More precisely, let $d\in \N_{\ge 1}, t\in\Z_{\geq 0}$ and $f(n)=dn+t$, we present the detailed Hausdorff dimension analysis of the set\begin{equation*} E_{f}(ψ):=\left\{x\in [0, 1): \sqrt[n]{a_{f(n)}(x)a_{2f(n)}(x)\cdots a_{nf(n)}(x)}\geq ψ(n) \ {\rm for \ infinitely \ many} \ n\in \N\right\}. \end{equation*} It is seen that the dimension is larger if $d$ is larger and $t$ has no contribution to the dimension.

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arXiv2023

Dual $p$-adic Diophantine approximation on manifolds

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $ψ$-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 approximation on arbitrary nondegenerate manifolds. The divergence part has also been resolved for the $p$-adic setting by Datta-Ghosh in 2022 for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open problem. In this paper, we prove the homogeneous $p$-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.

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arXiv2023

Weighted twisted inhomogeneous Diophantine approximation

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, $Ψ$ an $n$-tuple of monotonic decreasing functions, and let $W_{A}(Ψ)$ be the set of points that infinitely often lie in a $Ψ(q)$-neighbourhood of the sequence $\{Aq\}_{q\in\mathbb{N}}$. We prove that the set $ W_{A}(Ψ)$ has zero-full Lebesgue measure under convergent-divergent sum conditions with some mild assumptions on $A$ and the approximating functions $Ψ$. We also prove the Hausdorff dimension results for this set. Along with some geometric arguments, the main ingredients are weighted ubiquity and weighted mass transference principle introduced recently by Kleinbock & Wang (Adv. Math. 2023), and Wang & Wu (Math. Ann. 2021) respectively.

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