A localized uniformly Jarník set in continued fractions

• Autorzy: Yuanhong Chen , Yu Sun , Xiaojun Zhao
• Języki publikacji: EN

Abstrakty

EN

For any x ∈ [0,1], let [a₁(x), a₂(x),...] be its continued fraction expansion and $qₙ(x)_{n≥1}$ be the sequence of the denominators of its convergents. For any τ>0, we call
$U(τ) = {x ∈ [0,1): |x - pₙ(x)/qₙ(x)}| < (1/qₙ(x))^{τ+2}$ for n ∈ ℕ ultimately
a uniformly Jarník set, a collection of points which can be uniformly well approximated by its convergents eventually. In this paper, instead of a constant function of τ, we consider a localized version of the above set, namely
$U_{loc}(τ) = {x ∈ [0,1): |x - pₙ(x)/qₙ(x)| < (1/qₙ(x))^{τ(x)+2}$ for n ∈ ℕ ultimately,
where τ:[0,1] → ℝ⁺ is a continuous function. We call $U_{loc}(τ)$ a localized uniformly Jarník set, and determine its Hausdorff dimension.

Kategorie tematyczne

• 28A80: Fractals
• 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 167
• Numer: 3
• Strony: 267-280

Daty

wydano

2015

Twórcy

autor

Yuanhong Chen

• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, Hubei, P.R. China

autor

Yu Sun

• Faculty of Science, Jiangsu University, 212013 Zhenjiang, Jiangsu, P.R. China

autor

Xiaojun Zhao

• School of Economics, Peking University, 100871 Beijing, P.R. China

Identyfikatory

DOI

10.4064/aa167-3-5