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A localized uniformly Jarník set in continued fractions

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Chen Y., Sun Y., Zhao X.

Acta Arithmetica

2015

tom 167

nr 3

267-280

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.

Ograniczanie wyników

1 Acta Arithmetica

1 Chen Y.

1 Sun Y.

1 Zhao X.

1 2015

Wyniki wyszukiwania

A localized uniformly Jarník set in continued fractions