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Lulu Fang

Publications and source records attributed to Lulu Fang.

15 recordsLinked to original sources
arXiv2025

On fast Lyapunov spectra for Markov-R\'{e}nyi maps

In this paper, we study the multifractal analysis for Markov-R\'{e}nyi maps, which form a canonical class of piecewise differentiable interval maps, with countably many branches and may contain a parabolic fixed point simultaneously, and do not assume any distortion hypotheses. We develop a geometric approach, independent of thermodynamic formalism, to study the fast Lyapunov spectrum for Markov-R\'{e}nyi maps. Our study can be regarded as a refinement of the Lyapunov spectrum at infinity. We demonstrate that the fast Lyapunov spectrum is a piecewise constant function, possibly exhibiting a discontinuity at infinity. Our results extend the works in \cite[Theorem 1.1]{FLWW13}, \cite[Theorem 1.2]{LR}, and \cite[Theorem 1.2]{FSW} from the Gauss map to arbitrary Markov-R\'{e}nyi maps, and highlight several intrinsic differences between the fast Lyapunov spectrum and the classical Lyapunov spectrum. Moreover, we establish the upper and lower fast Lyapunov spectra for Markov-R\'{e}nyi maps.

math.DS↗
arXiv2024

Fractal geometry of continued fractions with large coefficients and dimension drop problems

In 1928, Jarn\'ık \cite{Jar} obtained that the set of continued fractions with bounded coefficients has Hausdorff dimension one. Good \cite{Goo} observed a dimension drop phenomenon by proving that the Hausdorff dimension of the set of continued fractions whose coefficients tend to infinity is one-half. For the set of continued fractions whose coefficients tend to infinity rapidly, Luczak \cite{Luc} and Feng et al. \cite{FWLT} showed that its Hausdorff dimension decreases even further. Recently, Liao and Rams \cite{LR16} also observed an analogous dimension drop phenomenon when they studied the subexponential growth rate of the sum of coefficients. In this paper, we consolidate and considerably extend the studies of the abovementioned problem into a general dimension drop problem on the distribution of continued fractions with large coefficients. As applications, we use a different approach to reprove a result of Wang and Wu on the dimensions of the Borel-Bernstein sets \cite{WW}, fulfil the dimension gap proposed by Liao and Rams \cite{LR16}, and establish several new results concerning the dimension theory of liminf and limsup sets related to the maximum of coefficients.

math.NT↗
arXiv2022

Dimensions of certain sets of continued fractions with non-decreasing partial quotients

Let $[a_1(x),a_2(x),a_3(x),\cdots]$ be the continued fraction expansion of $x\in (0,1)$. This paper is concerned with certain sets of continued fractions with non-decreasing partial quotients. As a main result, we obtain the Hausdorff dimension of the set \[\left\{x\in(0,1): a_1(x)\leq a_2(x)\leq \cdots,\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\}\] for any $ψ:\mathbb{N}\rightarrow\mathbb{R}^+$ satisfying $ψ(n)\to\infty$ as $n\to\infty$.

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arXiv2021

On upper and lower fast Knintchine spectra of continued fractions

Let $ψ:\mathbb{N}\to \mathbb{R}^+$ be a function satisfying $ϕ(n)/n\to \infty$ as $n \to \infty$. We investigate from a multifractal analysis point of view the growth rate of the sums $\sum^n_{k=1}\log a_k(x)$ relative to $ψ(n)$, where $[a_1(x),a_2(x), a_3(x)\cdots]$ denotes the continued fraction expansion of $x\in (0,1)$. The upper (resp. lower) fast Khintchine spectrum is defined by the Hausdorff dimension of the set of all points $x$ for which the upper (resp. lower) limit of $\frac{1}{ψ(n)}\sum^n_{k=1}\log a_k(x)$ is $1$. The precise formulas of these two spectra are completely determined, which strengthens a result of Liao and Rams (2016).

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arXiv2020

Precise asymptotics on the Birkhoff sums for dynamical systems

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdős, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in $(0,1)$.

math.DS↗
arXiv2020

Some exceptional sets of Borel-Bernstein Theorem in continued fractions

Let $[a_1(x),a_2(x), a_3(x),\cdots]$ denote the continued fraction expansion of a real number $x \in [0,1)$. This paper is concerned with certain exceptional sets of the Borel-Bernstein Theorem on the growth rate of $\{a_n(x)\}_{n\geq1}$. As a main result, the Hausdorff dimension of the set \[ E_{\sup}(ψ)=\left\{x\in[0,1):\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\} \] is determined, where $ψ:\mathbb{N}\rightarrow\mathbb{R}^+$ tends to infinity as $n\to\infty$.

math.NT↗
arXiv2018

A note on the run length function for intermittency maps

We study the run length function for intermittency maps. In particular, we show that the longest consecutive zero digits (resp. one digits) having a time window of polynomial (resp. logarithmic) length. Our proof is relatively elementary in the sense that it only relies on the classical Borel-Cantelli lemma and the polynomial decay of intermittency maps. Our results are compensational to the Erdős-Rényi law obtained by Denker and Nicol in \cite{dennic13}.

math.DS↗
arXiv2016

A remark on the extreme value theory for continued fractions

Let $x$ be a irrational number in the unit interval and denote by its continued fraction expansion $[a_1(x), a_2(x), \cdots, a_n(x), \cdots]$. For any $n \geq 1$, write $T_n(x) = \max_{1 \leq k \leq n}\{a_k(x)\}$. We are interested in the Hausdorff dimension of the fractal set \[ E_ϕ= \left\{x \in (0,1): \lim_{n \to \infty} \frac{T_n(x)}{ϕ(n)} =1\right\}, \] where $ϕ$ is a positive function defined on $\mathbb{N}$ with $ϕ(n) \to \infty$ as $n \to \infty$. Some partial results have been obtained by Wu and Xu, Liao and Rams, and Ma. In the present paper, we further study this topic when $ϕ(n)$ tends to infinity with a doubly exponential rate as $n$ goes to infinity.

math.NT↗
arXiv2016

The denominators of convergents for continued fractions

For any real number $x \in [0,1)$, we denote by $q_n(x)$ the denominator of the $n$-th convergent of the continued fraction expansion of $x$ $(n \in \mathbb{N})$. It is well-known that the Lebesgue measure of the set of points $x \in [0,1)$ for which $\log q_n(x)/n$ deviates away from $π^2/(12\log2)$ decays to zero as $n$ tends to infinity. In this paper, we study the rate of this decay by giving an upper bound and a lower bound. What is interesting is that the upper bound is closely related to the Hausdorff dimensions of the level sets for $\log q_n(x)/n$. As a consequence, we obtain a large deviation type result for $\log q_n(x)/n$, which indicates that the rate of this decay is exponential.

math.NT↗
arXiv2016

A note on Rényi's "record" problem and Engel's series

In 1973, Williams introduced two interesting discrete Markov processes, namely $C$-processes and $A$-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between $C$-processes and $A$-processes in the context of large deviations.

math.ST↗
arXiv2016

Approximation orders of real numbers by $β$-expansions

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $β$-expansions with the exponential order $β^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under $β$-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of $β$-expansions.

math.NT↗
arXiv2016

Beta-expansion and continued fraction expansion of real numbers

Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\logβ)/π^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\logβ)/π^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $β= 10$ to any $β>1$. Moreover, we also discuss which of the $β$-expansion and continued fraction expansion yields the better approximations of real numbers.

math.NT↗
arXiv2016

Limit theorems related to beta-expansion and continued fraction expansion

Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence $\{k_n, n \geq 1\}$, which generalize the results of Faivre and Wu respectively from $β=10$ to any $β>1$.

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arXiv2016

Random Continued fractions: Lévy constant and Chernoff-type estimate

Given a stochastic process $\{A_n, n \geq 1\}$ taking values in natural numbers, the random continued fractions is defined as $[A_1, A_2, \cdots, A_n, \cdots]$ analogue to the continued fraction expansion of real numbers. Assume that $\{A_n, n \geq 1\}$ is ergodic and the expectation $E(\log A_1) < \infty$, we give a Lévy-type metric theorem which covers that of real case presented by Lévy in 1929. Moreover, a corresponding Chernoff-type estimate is obtained under the conditions $\{A_n, n \geq 1\}$ is $ψ$-mixing and for each $0< t< 1$, $E(A_1^t) < \infty$.

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