The sets of Dirichlet non-improvable numbers versus well-approximable numbers

Bakhtawar, Ayreena; Bos, Philip J.; Hussain, Mumtaz

doi:10.1017/etds.2019.41

Cited by 23 publications

(16 citation statements)

References 16 publications

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“…It can readily be seen that the divergence case of Jarnik's theorem played a pivotal role in proving the divergence case of the Theorem 1.5. 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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Section: Continued Fractions and Dirichlet Improvabilitymentioning

confidence: 90%

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

Bos¹,

Hussain²,

Simmons³

2020

Preprint

Self Cite

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“…This is in turn related to the Dirichlet improvable sets discussed in [1,8,9]. This will be the subject of a forthcoming article.…”

Section: Introductionmentioning

confidence: 87%

“…They showed that if lim n→∞ ϕ(n) = ∞ and lim n→∞ ϕ(n)/ϕ(n + ϕ(n)) = 1, then the set E ϕ α,β is of full Hausdorff dimension for all α, β with 0 ≤ α ≤ β ≤ ∞. Tong et al [19] generalised these results to the βexpansion x = ∞ k=1 x n /β n for β ∈ (1,2]. (For more information on the β-expansion, see [2,6,22].…”

Section: Introductionmentioning

confidence: 99%

On the Longest Block Function in Continued Fractions

Song¹,

Zhou²

2020

Bull. Aust. Math. Soc.

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For an irrational number $x\in [0,1)$ , let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x):n\geq 1\}$ . Given $\unicode[STIX]{x1D6E9}\in \mathbb{N}$ , for $n\geq 1$ , the $n$ th longest block function of $x$ with respect to $\unicode[STIX]{x1D6E9}$ is defined by $L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$ , which represents the length of the longest consecutive sequence whose elements are all $\unicode[STIX]{x1D6E9}$ from the first $n$ partial quotients of $x$ . We consider the growth rate of $L_{n}(x,\unicode[STIX]{x1D6E9})$ as $n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.

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“…The result of Barral and Seuret was further generalized to the setting of continued fractions by Wang et al in [21]. To refer to their result, we first restate the Jarník-Besicovitch set (1)(2)(3)(4) in terms of growth rate of partial quotients, J(τ) = {x ∈ [0, 1) : a n (x) ≥ e ((τ−2)/2)S n (log |T (x)|) for i.m. n ∈ N}.…”

Section: Introductionmentioning

confidence: 99%

“…dim H {x ∈ [0, 1) : a n+1 (x) ≥ e τ(x)S n h(x) for i.m. n ∈ N} = s (1) N . • When r = 1, τ(x) = 1 and h(x) = log B, we obtain the result by Wang and Wu [20].…”

Section: Introductionmentioning

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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The sets of Dirichlet non-improvable numbers versus well-approximable numbers

Abstract: Let Ψ : [1, ∞) → R + be a non-decreasing function, a n

Cited by 23 publications

References 16 publications

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

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

On the Longest Block Function in Continued Fractions

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

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