BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Mumtaz Hussain (La Trobe University)
DTSTART:20200605T040000Z
DTEND:20200605T043000Z
DTSTAMP:20260417T112111Z
UID:NTOC2020/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTOC2020/13/
 ">Uniform Diophantine approximation: improving Dirichlet’s theorem</a>\n
 by Mumtaz Hussain (La Trobe University) as part of Number Theory Online Co
 nference 2020\n\n\nAbstract\nIn this talk\, I will discuss the metrical th
 eory associated with the set of Dirichlet non-improvable numbers. \n\nLet 
 $\\Psi :[1\,\\infty )\\rightarrow \\mathbb{R}_{+}$ be a non-decreasing fun
 ction\, $a_{n}(x)$ the $n$'th partial quotient of $x$ and $q_{n}(x)$ the d
 enominator of the $n$'th convergent. The set of $\\Psi $-Dirichlet non-imp
 rovable numbers \n$$\nG(\\Psi):=\\Big\\{x\\in \\lbrack 0\,1):a_{n}(x)a_{n+
 1}(x)\\\,>\\\,\\Psi \\big(q_{n}(x)\n\\big)\\ \\mathrm{for\\ infinitely\\ m
 any}\\ n\\in \\mathbb{N}\\Big\\}\,\n$$\nis related with the classical set 
 of $1/q^{2}\\Psi (q)$-approximable numbers \n\n$$\n\\mathcal{K}(\\Psi):=\\
 left\\{x\\in[0\,1): \\left|x-\\frac pq\\right|<\\frac{1}{\nq^2\\Psi(q)} \\
  \\mathrm{for \\ infinitely \\ many \\ } (p\, q)\\in \\mathbb{Z}\\times \\
 mathbb{N }\n\\right\\}\,\n$$\n in the sense that $\\mathcal{K}(3\\Psi )\\s
 ubset G(\\Psi )$. In this talk\, I will explain that the Hausdorff measure
  of the set  $G(\\Psi)$ obeys a zero-infinity law for a large class of dim
 ension functions.  Together with the Lebesgue measure-theoretic results es
 tablished by Kleinbock \\& Wadleigh (2016)\, our results contribute to bui
 lding a complete metric theory for the set of Dirichlet non-improvable num
 bers.\n\nAnother recent result that I will discuss will be the Hausdorff d
 imension of the set  $G(\\Psi)\\setminus \\mathcal{K}(3\\Psi )$.\n
LOCATION:https://researchseminars.org/talk/NTOC2020/13/
END:VEVENT
END:VCALENDAR