On digit frequencies in 𝛽-expansions

Boyland, Philip; Carvalho, André de; Hall, Toby

doi:10.1090/tran/6617

Cited by 10 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As can be seen in Figure 2, γ(β) is a devil's staircase: it is continuous, non-decreasing, and has zero derivative almost everywhere. The same function arises when considering digit frequencies for β-expansions, as described by Boyland et al in [1].…”

Section: 1mentioning

confidence: 79%

The $\beta$-transformation with a hole

Clark¹

2015

DCDS-A

View full text Add to dashboard Cite

This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_{\beta}(x)=\beta x \pmod 1$. Let $\mathcal{J}_{\beta} (a,b) := \{ x \in (0,1) : T_{\beta}^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every $n$-cycle for $T_{\beta}$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\beta(a,b)$. In this paper we completely describe the following sets: \[ D_0(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \neq \emptyset \}, \] \[ D_1(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \text{ is uncountable} \}, \] \[ D_2(\beta) = \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}. \]Comment: 20 pages, 8 figures. Updated version has added examples and a small mistake in Lemma 3.4 has been fixe

show abstract

Section: 1mentioning

confidence: 79%

The $\beta$-transformation with a hole

Clark¹

2015

DCDS-A

View full text Add to dashboard Cite

show abstract

“…Since then, much attention has been paid to the β-dynamical system ([0, 1], T β , ν β ) and β-expansions of real numbers, see [3][4][5][6][7][8], etc, and references therein. Given β > 1, for any x ∈ [0, 1], the sequence ε(x, β) = ε 1 (x, β)ε 2 (x, β) .…”

Section: Introductionmentioning

confidence: 99%

Maximal run-length function for real numbers in beta-dynamical system

Lü

2020

Nonlinearity

View full text Add to dashboard Cite

Let β > 1 and x ∈ [0, 1) be two real numbers. For any y ∈ [0, 1), the maximal run-length function r x (y, n) (with respect to x) is de ned to be the maximal length of the pre x of x's β-expansion which appears in the rst n digits of y's.In this paper, we study the metric properties of the maximal run-length function and apply them to the hitting time, which generalises many known results. In the meantime, the fractal dimensions of the related exceptional sets are also determined.

show abstract

“…Despite being a subject that has its origins in the early 20th century, representations of real numbers and their digit frequencies is still motivating researchers. For some recent contributions in this area see [9,13,16] and the references therein. Most of the existing work in this area was done in a setting where the representation is unique.…”

Section: Introductionmentioning

confidence: 99%

Exceptional digit frequencies and expansions in non-integer bases

Baker

2019

Monatsh Math

View full text Add to dashboard Cite

In this paper we study the set of digit frequencies that are realised by elements of the set of β-expansions. The main result of this paper demonstrates that as β approaches 1, the set of digit frequencies that occur amongst the set of β-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.

show abstract

On digit frequencies in 𝛽-expansions

Cited by 10 publications

References 20 publications

The $\beta$-transformation with a hole

The $\beta$-transformation with a hole

Maximal run-length function for real numbers in beta-dynamical system

Exceptional digit frequencies and expansions in non-integer bases

Contact Info

Product

Resources

About