Metrical theory for the set of points associated with the generalized Jarnik-Besicovitch set

Ayreena Bakhtawar (La Trobe University)

20-Apr-2021, 12:30-13:30 (3 years ago)

Abstract: From Lagrange's (1770) and Legendre's (1808) results we conclude that to find good rational approximations 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 fraction 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 quotient of $x$ and $T$ is the Gauss map. In this talk, I will focus on determining the Hausdorff dimension of the set of real numbers $x \in [0,1)$ such that for any $m \in \mathbb{N}$ the following holds for infinitely many $n \in \mathbb{N}$: $a_{n+1}(x) a_{n+2}(x) \cdots a_{n+m}(x) \ge e^{τ(x)\, (f(x)+⋯+f(T^{n-1}x))}$, 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 including the famous Jarnik-Besicovitch theorem.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )


One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

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Organizers: Shigeki Akiyama, Ayreena Bakhtawar, Karma Dajani, Kevin Hare, Hajime Kaneko, Niels Langeveld, Lingmin Liao, Wolfgang Steiner*
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