The shrinking target problem in the dynamical system of continued fractions
speaker: Bing Li (Department of Mathematical Sciences, University of Oulu, Finland)abstract:
Let \(([0,1), T)\) be the dynamical system of continued fractions. Let \(\{z_n\}_{n\ge 1}\) be a sequence of real numbers in \([0,1]\) and \(\psi: \mathbb{N}\times [0,1)\to \mathbb{R}^+\) be a positive function. A point \(x\in [0,1)\) is said to be \(\psi\)-approximable by \(\{z_n\}_{n\ge 1}\) if \(
T^nx-z_n
<\psi(n,x)\) holds for infinitely many \(n\in \mathbb{N}\). In this paper, the Hausdorff dimension of the set of \(\psi\)-approximable points is studied. The dimensions are completely determined when \(\psi(n,x)=\psi(n)\) independent on \(x\) and when \(\psi(n,x)=e^{-(f(x)+\cdots+f(T^{n-1}x))}\) with \(f\) a positive continuous function. For the proof of these results, a relationship between a ball in \([0,1)\) and the cylinders defined by the partial quotients in continued fractions is investigated. It is shown that a ball can be sufficiently packed by cylinders
of the same order and of comparable length, which gives us explicit continued fraction representations in locating the points in a ball in \([0,1)\).
timetable:
Tue 11 Jun, 17:00 - 17:25, Aula Dini
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