Metric properties of the product of consecutive partial quotients in continued fractions

Huang, Lingling; Wu, Jun; Xu, Jian

doi:10.1007/s11856-020-2049-1

Cited by 24 publications

(29 citation statements)

References 24 publications

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“…The work of Kleinbock-Wadleigh was followed by Huang-Wu-Xu [12] with both Lebesgue measure and Hausdorff dimension results for a natural generalization of the set (1.2). Namely, for m ∈ N they considered…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…This lemma can be observed from the positions of the cylinders in Proposition 2.3. A detailed proof can be found in [12].…”

Section: Cantor Structure Of Ementioning

confidence: 99%

“…E m (Ψ) := {x ∈ [0, 1) : a n (x) • • • a n+m−1 (x) ≥ Ψ(n) for infinitely many n ∈ N} ,and proved the following Theorem 1.3 (Huang-Wu-Xu, 2019). Given Ψ : N → R ≥1 , (a)[12, Theorem 1.5] …”

mentioning

confidence: 99%

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Metrical properties for the weighted products of multiple partial quotients in continued fractions

Bakhtawar¹,

Hussain²,

Kleinbock³

et al. 2022

Preprint

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The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set:where t i ∈ R + for all 0 ≤ i ≤ m − 1, and Ψ : N → R ≥1 is a positive function.

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Section: Statement Of Resultsmentioning

confidence: 99%

“…This lemma can be observed from the positions of the cylinders in Proposition 2.3. A detailed proof can be found in [12].…”

Section: Cantor Structure Of Ementioning

confidence: 99%

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Metrical properties for the weighted products of multiple partial quotients in continued fractions

Bakhtawar¹,

Hussain²,

Kleinbock³

et al. 2022

Preprint

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“…Further, it is worth pointing out that the difference set G(Ψ) \ K(3Ψ) is non-trivial as the Hausdorff dimension of this set is the same as that of G(ψ) as proved in [1]. We refer the reader to [7] for the Lebesgue measure and Hausdorff dimension results for a generalised form of the set G(Ψ).…”

Section: Continued Fractions and Dirichlet Improvabilitymentioning

confidence: 90%

The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Bos¹,

Hussain²,

Simmons³

2020

Preprint

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Let ψ : R + → R + be a non-increasing function. A real number x is said to be ψ-Dirichlet improvable if the system |qx − p| < ψ(t) and |q| < t has a non-trivial integer solution for all large enough t. Denote the collection of such points by D(ψ). Hussain-Kleinbock-Wadleigh-Wang (2018) proved the Hausdorff measure of the set D c (ψ) (the set of ψ-Dirichlet non-improvable numbers) by showing that it obeys a zero-infinity law for a large class of (essentially sub-linear) dimension functions. In this paper, we prove the generalised Hausdorff f -measure of the set D c (ψ) for non essentially sub-linear dimension functions. Hence, together with the Hussain-Kleinbock-Wadleigh-Wang (2018) result, we provide a complete characterisation of the generalised Hausdorff f -measure of the set D c (ψ).

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“…The limit defining P B (T, ψ) exists. Furthermore, if ψ : [0, 1) → R is a function satisfying the tempered distortion property, then the value of P B (T, ψ) remains unchanged even without taking the supremum over x ∈ Z B in (3)(4)(5)(6)(7)(8)(9)(10). The independence of P B (T, ψ) on the supremum over x ∈ Z B follows from the fact that…”

mentioning

confidence: 99%

Hausdorff Dimension for the Set of Points Connected With the Generalized Jarník–besicovitch Set

Bakhtawar

2020

J. Aust. Math. Soc.

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In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$ , $$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$ holds for infinitely many $n\in \mathbb {N}$ , where h and $\tau $ are positive continuous functions, T is the Gauss map and $a_{n}(x)$ denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of $r,\tau (x)$ and $h(x)$ we obtain various classical results including the famous Jarník–Besicovitch theorem.

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Metric properties of the product of consecutive partial quotients in continued fractions

Cited by 24 publications

References 24 publications

Metrical properties for the weighted products of multiple partial quotients in continued fractions

Metrical properties for the weighted products of multiple partial quotients in continued fractions

The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Hausdorff Dimension for the Set of Points Connected With the Generalized Jarník–besicovitch Set

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