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• # Artykuł - szczegóły

## Acta Arithmetica

2015 | 167 | 3 | 267-280

## A localized uniformly Jarník set in continued fractions

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.

267-280

wydano
2015

### Twórcy

autor
• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, Hubei, P.R. China
autor
• Faculty of Science, Jiangsu University, 212013 Zhenjiang, Jiangsu, P.R. China
autor
• School of Economics, Peking University, 100871 Beijing, P.R. China