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- Title
Dimension theory of the product of partial quotients in Lüroth expansions.
- Authors
Tan, Bo; Zhou, Qinglong
- Abstract
For x ∈ [ 0 , 1) , let [ d 1 (x) , d 2 (x) , ... ] be its Lüroth expansion and { p n (x) q n (x) , n ≥ 1 } be the sequence of convergents of x. Define the sets 𝜀 2 (φ) = { x ∈ [ 0 , 1) : d n + 1 (x) d n (x) ≥ φ (n) for infinitely many n ∈ ℕ } , U ∗ (τ) = x ∈ [ 0 , 1) : x − p n (x) q n (x) < 1 q n (x) (τ + 1) for n ∈ ℕ ultimately and F (τ) = x ∈ [ 0 , 1) : lim n → ∞ log (d n (x) d n + 1 (x)) log q n (x) = τ , where φ : ℕ → [ 2 , ∞) is a positive function. In this paper, we calculate the Lebesgue measure of the set 𝜀 2 (φ) and the Hausdorff dimension of the sets U ∗ (τ) and F (τ).
- Subjects
FRACTAL dimensions; LEBESGUE measure; DIOPHANTINE approximation
- Publication
International Journal of Number Theory, 2021, Vol 17, Issue 5, p1139
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042121500287