BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ayreena Bakhtawar (La Trobe University) DTSTART:20210420T123000Z DTEND:20210420T133000Z DTSTAMP:20260423T052922Z UID:OWNS/41 DESCRIPTION:Title: Me trical theory for the set of points associated with the generalized Jarnik -Besicovitch set\nby Ayreena Bakhtawar (La Trobe University) as part o f One World Numeration seminar\n\n\nAbstract\nFrom Lagrange's (1770) and L egendre's (1808) results we conclude that to find good rational approximat ions to an irrational number we only need to focus on its convergents. Let $[a_1(x)\,a_2(x)\,\\dots]$ be the continued fraction expansion of a real number $x \\in [0\,1)$. The Jarnik-Besicovitch set in terms of continued f raction consists of all those $x \\in [0\,1)$ which satisfy $a_{n+1}(x) \\ ge e^{\\tau\\\, (\\log|T'x|+⋯+\\log|T'(T^{n-1}x)|)}$ for infinitely many $n \\in \\mathbb{N}$\, where $a_{n+1}(x)$ is the $(n+1)$-th partial quoti ent of $x$ and $T$ is the Gauss map. In this talk\, I will focus on determ ining the Hausdorff dimension of the set of real numbers $x \\in [0\,1)$ s uch that for any $m \\in \\mathbb{N}$ the following holds for infinitely m any $n \\in \\mathbb{N}$: $a_{n+1}(x) a_{n+2}(x) \\cdots a_{n+m}(x) \\ge e ^{τ(x)\\\, (f(x)+⋯+f(T^{n-1}x))}$\, where $f$ and $\\tau$ are positive continuous functions. Also we will see that for appropriate choices of $m$ \, $\\tau(x)$ and $f(x)$ our result implies various classical results incl uding the famous Jarnik-Besicovitch theorem.\n LOCATION:https://researchseminars.org/talk/OWNS/41/ END:VEVENT END:VCALENDAR