arXiv subjects

Mumtaz Hussain

Publications and source records attributed to Mumtaz Hussain.

At least 37 records · Page 2Linked to original sources

arXiv2023

Metrical properties of weighted products of consecutive Lüroth digits

The Lüroth 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 $Ψ:\mathbb{N}\to (1,\infty)$, define \[ \mathcal{E}_{\mathbf{t}}(Ψ)\colon= \left\{ x\in (0,1]: d_n^{t_0} \cdots d_{n+m}^{t_{m-1}}\geq Ψ(n) \text{ for infinitely many} \ n \in\mathbb{N} \right\}. \] We establish a Lebesgue measure dichotomy statement (a zero-one law) for $\mathcal{E}_{\mathbf{t}}(Ψ)$ under a natural non-removable condition $\liminf_{n\to\infty} Ψ(n)>~1$. Let $B$ be given by \[ \log B \colon= \liminf_{n\to\infty} \frac{\log(Ψ(n))}{n}. \] For any $m\in\mathbb{N}$, we compute the Hausdorff dimension of $\mathcal{E}_{\mathbf{t}}(Ψ)$ when either $B=1$ or $B=\infty$. We also compute the Hausdorff dimension of $\mathcal{E}_{\mathbf{t}}(Ψ)$ when $1<B< \infty$ for $m=2$.

math.NT↗

arXiv2023

An exponentially shrinking problem

The Jarník-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 $h>0, τ\geq 1$, and for any $j\geq 1$ define the integer sequence $q_{j+1}=q_j^h$. We prove the Hausdorff dimension of the set $$Λ^\bftheta_d(τ)=\left\{\xx\in[0, 1)^d: \|q_jx_i-θ_i\|<q_j^{-τ} \ \text{for all } j\geq 1, i=1,2,\cdots,d\right\},$$ where $\left\|\star\right\|$ denotes the distance to the nearest integer and $\bftheta\in [0, 1)^d$ is fixed. We also give some heuristics for the Hausdorff dimension of the corresponding multiplicative set $$\MM_d^\bftheta(τ)=\left\{\xx\in[0, 1)^d:\prod_{i=1}^d \|q_jx_i-θ_i\|<q_j^{-τ} \ \text{for all } j\geq 1\right\}.$$

math.NT↗

arXiv2023

The Baker-Schmidt problem for dual approximation and some classes of functions

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff $f$-measure of the set of $Ψ$-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 classes of nondegenerate submanifolds of codimension greater than $1$. Concretely, for codimension two or three, we provide examples of manifolds where the dependent variables can be chosen as quadratic forms. Our method requires the manifold to have even dimension at least the minimum of four and half the dimension of the ambient space. We conjecture that these restrictions on the dimension of the manifold are sufficient to provide similar examples in general.

math.NT↗

arXiv2022

The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Let $ψ:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $ψ$-Dirichlet improvable if the system $$|qx-p|< \, ψ(t) \ \ {\text{and}} \ \ |q|<t$$ 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 non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sub-linear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang (2018), for some non-essentially sub-linear dimension functions, and for all dimension functions but with a growth condition on the approximating function.

math.NT↗

arXiv2022

A Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions

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(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions such that $\lim\limits_{n\to\infty} Φ_i(n)=\infty$. We obtain some surprising results including the situations when $\FF(Φ_1,Φ_2)$ is empty for various non-trivial choices of $Φ_i$'s. Our results contribute to the metrical theory of continued fractions by generalising several known results including the main result of [Nonlinearity, 33(6):2615--2639, 2020]. To obtain some of the results, we consider an alternate generalised set, which may be of independent interest, and calculate its Hausdorff dimension. One of the main ingredients is in the usage of the classical mass distribution principle; specifically a careful distribution of the mass on the Cantor subset by introducing a new idea of two different types of probability measures.

math.NT↗

arXiv2022

Diophantine approximation on curves

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 $Ψ$-approximable points on nondegenerate manifolds. The problem relates the `size' of the set of $Ψ$-approximable points with the convergence or divergence of a certain series. In the dual approximation setting, the divergence case has been established by Beresnevich-Dickinson-Velani (2006) for any nondegenerate manifold. The convergence case, however, represents a major challenging open problem and progress thus far has been effectuated in limited cases only. In this paper, we discuss and prove several results on the $\HH^g$-measure on Veronese curves in any dimension $n$. As a consequence of one of our results, we generalize recent results of Pezzoni [Acta Arith. 193 (2020), no. 3, 269-281] regarding $n=3$. This improvement evolves from a deeper investigation on general irreducibility considerations applicable in arbitrary dimensions. We further investigate the $\HH^g$-measure for convergence on planar curves. We show that the monotonicity assumption on a multivariable approximating function cannot be removed for planar curves.

math.NT↗

arXiv2022

Higher dimensional shrinking target problem in beta dynamical systems

We consider the two dimensional shrinking target problem in the beta dynamical system for general $β>1$ and with the general error of approximations. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in[0,1]$, define the shrinking target set $$ E(T_β, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{ll} |T_β^{n}x-x_{0}|<e^{-S_nf(x)}\\ [1ex] |T_β^{n}y-y_{0}|< e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $S_nf(x)=\sum_{j=0}^{n-1}f(T_β^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher dimensional beta dynamical systems.

math.NT↗

arXiv2022

Metrical properties for continued fractions of formal Laurent series

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 consider the sets of $x$ such that $°A_{n+1}(x)+\cdots+°A_{n+k}(x)~\ge~Φ(n)$ holds for infinitely many $n$ and for all $n$ respectively, where $k\ge1$ is an integer and $Φ(n)$ is a positive function defined on $\mathbb{N}$. We determine the size of these sets in terms of Haar measure and Hausdorff dimension.

math.NT↗

arXiv2022

Metrical theorems for unconventional height functions

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).

math.NT↗

arXiv2022

Limit theorems for sums of products of consecutive partial quotients of continued fractions

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 partial sums $S_n(x)= \sum_{i=1}^n a_i(x)a_{i+1}(x)$ as well as, from a multifractal analysis point of view, investigate its increasing rate. Specifically, we prove the following results: \medskip \begin{itemize} \item For any $ε>0$, the Lebesgue measure of the set $$\left\{x\in(0, 1): \left|\frac{ S_n(x)}{n\log^2 n}-\frac1{2\log2}\right|\geq ε\right\}$$tends to zero as $n$ to infinity. \item For Lebesgue almost all $x\in (0,1)$, $$\lim\limits_{n\rightarrow \infty} \frac{S_n(x)-\max\limits_{1\leq i \leq n}a_i(x)a_{i+1}(x)}{n\log^2n}=\frac{1}{2\log2}.$$ \item The Hausdorff dimension of the set $$E(ϕ):=\left\{x\in(0,1):\lim\limits_{n\rightarrow \infty}\frac{S_n(x)}{ϕ(n)}=1\right\}$$ is determined for a range of increasing functions $ϕ: \mathbb N\to \mathbb R^+$. \end{itemize}

math.NT↗

arXiv2022

Metrical properties for the weighted products of multiple partial quotients in continued fractions

The classical Khintchine and Jarník 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 it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set: $$ {\D_{\mathbf t}}(ψ):=\left\{x\in[0, 1): \prod\limits_{i=0}^{m-1}{a^{t_i}_{n+i}(x)} \ge Ψ(n)\ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $t_i\in\mathbb R_+$ for all ${0\leq i\leq m-1}$, and $Ψ:\N\to\R_{\ge 1}$ is a positive function.

math.NT↗

arXiv2020

Dynamical Borel-Cantelli lemma for recurrence theory

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, μ, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $μ$-measure of the following set \[ R(ψ)= \{x\in X : d(T^n x, x) < ψ(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $ψ:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.

math.DS↗

arXiv2020

Metrical theorems on systems of affine forms

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 $\qq\mapsto \qq X+\bfalpha$, where $\qq\in\Z^m$ (viewed as a row vector), $X$ is an $m\times n$ real matrix and $\bfalpha\in \R^n$. The classical setting refers to the ${\rm dist}(\qq X+\bfalpha, \Z^m)$ to measure the closeness of the integer values of the system $(X, \bfalpha)$ to integers. The absolute value setting is obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, \0)$; and the more general mixed settings are obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, Λ)$, where $Λ$ is a subgroup of $\Z^m$. We prove the Khintchine--Groshev and Jarník type theorems for the mixed affine forms and Jarník type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).

math.NT↗

arXiv2019

The generalised Baker--Schmidt problem on hypersurfaces

The Generalised Baker--Schmidt Problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $ψ$-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--Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The corresponding convergence counterpart represents a major challenging open question and the progress thus far has only been attained over planar curves. In this paper, we settle this problem for hypersurfaces in a more general setting, i.e. for inhomogeneous approximations and with a non-monotonic multivariable approximating function.

math.NT↗

arXiv2019

Hausdorff dimension of a set in the theory of continued fractions

In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $Φ:\mathbb{N}\rightarrow (1,\infty)$ is any function with $\lim_{n\to \infty} Φ(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.

math.DS↗

arXiv2019

The sets of Dirichlet non-improvable numbers vs well-approximable numbers

Let $Ψ:[1,\infty )\rightarrow \mathbb{R}_{+}$ 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 $Ψ$-Dirichlet non-improvable numbers \begin{equation*} G(Ψ):=\Big\{x\in \lbrack 0,1):a_{n}(x)a_{n+1}(x)\,>\,Ψ\big(q_{n}(x) \big)\ \mathrm{for\ infinitely\ many}\ n\in \mathbb{N}\Big\}, \end{equation*} is related with the classical set of $1/q^{2}Ψ(q)$-approximable numbers $ \mathcal{K}(Ψ)$ in the sense that $\mathcal{K}(3Ψ)\subset G(Ψ)$. Both of these sets enjoy the same $s$-dimensional Hausdorff measure criterion for $s\in (0,1)$. We prove that the set $G(Ψ)\setminus \mathcal{K}(3Ψ)$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets $\mathcal{K}(Ψ)$ and $G(Ψ)$. This gives an affirmative answer to a question raised by Hussain-Kleinbock-Wadleigh-Wang (2018).

math.NT↗

arXiv2019

A general principle for Hausdorff measure

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).

math.MG↗

arXiv2018

A problem in non-linear Diophantine approximation

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 associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.

math.NT↗