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pro vyhledávání: '"Hussain, Mumtaz"'
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 $\varphi:\mathbb{N}\to\mathbb{R}_{>0}$ and $\ell\in \mathbb{
Externí odkaz:
http://arxiv.org/abs/2405.10538
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
Externí odkaz:
http://arxiv.org/abs/2402.13451
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}): a_n(x)\in D \text{
Externí odkaz:
http://arxiv.org/abs/2312.17388
Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian rational. However, the resulting space of sequences of Gaussian integers $\Omega$ is not closed. In this paper, we show th
Externí odkaz:
http://arxiv.org/abs/2310.20029
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 $\tau\geq 2$ and any non-empty proper subset $J(b)$ o
Externí odkaz:
http://arxiv.org/abs/2310.11698
Autor:
Hussain, Mumtaz, Ward, Ben
The Jarn\'ik-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=
Externí odkaz:
http://arxiv.org/abs/2309.13338
Autor:
Hussain, Mumtaz, Shulga, Nikita
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 $\Phi$, Wa
Externí odkaz:
http://arxiv.org/abs/2309.10529
Autor:
Hussain, Mumtaz, Shulga, Nikita
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 $\Phi:\mathbb N\to [2,+\infty)$ and any integer $d\geq 1$, we det
Externí odkaz:
http://arxiv.org/abs/2309.00826
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
Externí odkaz:
http://arxiv.org/abs/2309.00353
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 weig
Externí odkaz:
http://arxiv.org/abs/2308.16603