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Mumtaz Hussain

Publications and source records attributed to Mumtaz Hussain.

48 records · Page 3Linked to original sources
arXiv2017

The Hausdorff measure version of Gallagher's theorem -- closing the gap and beyond

In this paper we prove an upper bound on the "size" of the set of multiplicatively $ψ$-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 series in both the homogeneous and inhomogeneous settings. This zero-full law resolves a question posed by Beresnevich and Velani (2015) regarding the "log factor" discrepancy in the convergent/divergent sum conditions of their theorem. We further prove the analogous result for the multiplicative doubly metric setup.

math.NT↗
arXiv2017

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 it admits an improvement to Dirichlet's theorem in the following sense: 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 that the Hausdorff measure of the complement $D(ψ)^c$ (the set of $ψ$-Dirichlet non-improvable numbers) obeys a zero-infinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock \& Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

math.NT↗
arXiv2016

A note on badly approximabe sets in projective space

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\'ık-type result also holds for `badly approximable' points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine's theorem for convergence relating to `intrinsic' approximation of points in these compact sets.

math.NT↗
arXiv2016

A dichotomy law for the Diophantine properties in $β$-dynamical systems

Let $β>1$ be a real number and define the $β$-transformation on $[0,1]$ by $T_β:x\mapsto βx\bmod 1$. Further, define $$W_y(T_β,Ψ):=\{x\in [0, 1]:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\}$$ and $$W(T_β,Ψ):=\{(x, y)\in [0, 1]^2:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\},$$ where $Ψ:\mathbb{N}\to\mathbb{R}_{>0}$ is a positive function such that $Ψ(n)\to 0$ as $n\to \infty$. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

math.NT↗
arXiv2015

An Inhomogeneous Jarník type theorem for planar curves

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 Jarník are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarník type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves and in so doing complete the metric theory for both the homogeneous and inhomogeneous settings for approximation on planar curves.

math.NT↗
arXiv2014

Layered Interference Alignment: Achieving the Total DoF of MIMO X Channels

The $K\times 2$ and $2\times K$, Multiple-Input Multiple-Output (MIMO) X channel with constant channel coefficients available at all transmitters and receivers is considered. A new alignment scheme, named \emph{layered interference alignment}, is proposed in which both vector and real interference alignment are exploited, in conjunction with joint processing at receiver sides. Data streams with fractional multiplexing gains are sent in the desired directions to align the interfering signals at receivers. To decode the intended messages at receivers, a joint processing/simultaneous decoding technique, which exploits the availability of several receive antennas, is proposed. This analysis is subsequently backed up by metrical results for systems of linear forms. In particular, for such linear forms, Khintchine--Groshev type theorems are proved over real and complex numbers. It is observed that $K\times 2$ and $2\times K$, X channels with $M$ antennas at all transmitters/receivers enjoy duality in Degrees of Freedom (DoF). It is shown that incorporating the layered interference alignment is essential to characterize the total DoF of $\frac{2KM}{K+1}$ in the $K\times 2$ and $2\times K$, $M$ antenna X channels.

cs.IT↗
arXiv2014

A Jarník type theorem for planar curves: everything about the parabola

The well known theorems of Khintchine and Jarník 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 more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.

math.NT↗
arXiv2013

A converse to linear independence criteria, valid almost everywhere

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.

math.NT↗
arXiv2012

On weighted inhomogeneous Diophantine approximation on planar curves

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 Hausdor? dimension results for the sets of simultaneously well-approximable points on planar curves, established in [2, 6, 10, 20].

math.NT↗
arXiv2011

Metrical results on systems of small linear forms

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 approximating function.

math.NT↗
arXiv2009

The Metrical Theory of Simultaneously Small Linear Forms

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 complete Hausdorff dimension theory.

math.NT↗