Daniel Ingebretson
We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued fractions. These bounds are solutions to Moran-type equations in the convergents that can be easily implemented in a computer algebra system.
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A problem of Hirst for the Hurwitz continued fraction and the Hausdorff dimension of sets with restricted slowly growing digits
针对复无理数的Hurwitz连分数数字受限且满足增长条件的集合，本文证实了Hirst猜想，建立了Wang和Wu正则连分数结果的复类比，还证明了第二作者受限慢增长数字集维数结果的复类比，均借助无限共形迭代函数系统。
Hausdorff dimension of sets of continued fractions with bounded odd and even order partial quotients
研究奇数阶部分商为1的连分数集合的大小，证明两个此类集合的和与积均包含非空区间，并确定奇数阶部分商为1且偶数阶部分商以指数、超指数及一般正函数速率增长时集合的Hausdorff维数。
Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions
本文研究了偏商序列严格递增且包含任意长等差数列的无理数的豪斯多夫维数。
In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $Φ:\mathbb{N}\rightarrow (1,\infty)$ is any function with $\lim_{n\to \infty} Φ(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.
Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions
本研究确定了半正则连分数中部分商受限且缓慢增长的无理数集合的豪斯多夫维数，推广了正则连分数的相关结果，通过构造非自治迭代函数系统并应用Bowen公式证明。
Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence
研究沿给定子序列的部分商趋于无穷的连分数集合的Hausdorff维数，证明维数由子序列的上密度决定：上密度为正时维数为1/2，上密度为零时维数为1。
We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set $E_2$ (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 deci…
Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals
研究了Jarník集与迪奥菲内分数集合交集的Hausdorff维度，证明其为满维，并指出无理数指数大于2且部分商有界的集合维度为1。
We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions…
Exact dimensions of the prime continued fraction Cantor set
本文研究了素数Cantor集的Hausdorff维数和packing维数，证明了其Hausdorff测度不能有限正，但在数论猜想下其packing测度关于特定维数函数正且有限。
Hausdorff Dimension of Growth Rate Level Sets in $θ$-expansions
研究θ展开中由数字增长速率定义的水平集的Hausdorff维度，证明对于任意α≥0，该集具有满维性，扩展了Zhang和Lü(2016)在常规连分数中的工作。
We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are given by a modified variational principle.
We provide new similarities between regular continued fractions and Lüroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the Lüroth transformation of results by W. Liu, B. Li and W. Liu, S. Wang on the distal, asymptotic and Li-Yorke pairs for the Gauss map.
Prescribed realisation of longest runs in continued fractions
研究连分数展开中固定符号游程与整体最长游程的相互作用，构造了具有完全Hausdorff维数的无理数集，使得规定值的最长游程具有指定的渐近增长，并唯一实现整体最大值。
For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(ψ)$ of complex numbers which are well approximated with the given bound $ψ$ and have quite different Hurwitz contin…
Good's Theorem for regular continued fraction states that the set of real numbers $[a_0;a_1,a_2,\ldots]$ such that $\displaystyle\lim_{n\to\infty} a_n=\infty$ has Hausdorff dimension $\tfrac{1}{2}$. We show an analogous result for the complex plane and Hurwitz Continued Fractions. The set of complex numbers whose Hurwitz Continued fraction $[a_0;a_1,a_2,\ldots]$ satisfies $\displaystyle\lim_{n\to\infty} |a_n|=\infty$ has Hausdorff dimension $1$, half of the ambient space's dimension.
It is known that nonergodic directions in a rational billiard form a subset of the unit circle with Hausdorff dimension at most 1/2. Explicit examples realizing the dimension 1/2 are constructed using Diophantine numbers and continued fractions. A lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction.
How many points contain homothetic copies in their Hurwitz continued fraction expansion?
研究证明复数无理数在Hurwitz连续分数展开中存在无限多个相似复制，其Hausdorff维度为1，揭示了数论展开中的多维模式形成。
To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fractio…
In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A.N. Starkov. As a by-product of the analysis, we obtain a higher dimensional generalisation of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the techniques used to compute Hausdorff dimension, we show that the set of real numbers with divergent partial quotients has Hausdorff…
Fractal geometry of continued fractions with large coefficients and dimension drop problems
本文将连分数大系数的各类维数下降现象统一为一般维数下降问题，并据此重新证明Borel-Bernstein集合维数结果、填补Liao-Rams维数间隙，同时建立与系数最大值相关的liminf和limsup集合维数新结论。
Continued fractions with large prime partial quotients
本文研究部分商为大素数的连分数实数集合，通过证明几乎素数zeta函数尾端的新渐近结果，确定了相应集合的勒贝格测度与豪斯多夫维数，推广并涵盖了多个已有经典与近期定理。
Metrical properties of the product of partial quotients with geometric mean in continued fractions
本文研究连分数中部分商乘积的几何平均的度量性质，分析了当指数呈线性间隔且乘积长度增长时，豪斯多夫维数的变化，发现维数随 $d$ 增大而增大，$t$ 不影响维数。
Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let $A_n(x)$ be the $n$th partial quotient of the continued fraction expansion of $x$ in the field of formal Laurent series. We consider the sets of $x$ such that $°A_{n+1}(x)+\cdots+°A_{n+k}(x)~\ge~Φ(n)$ holds for infinitely many $n$ and for all $n$ respectively, where $k\ge1$ is an integer and $Φ(n)$ is a positive function defined on $\mathbb{N}$. We determine the size of these…
For an irrational number $x\in [0,1)$, let $x=[a\_1(x), a\_2(x),\cdots]$ be its continued fraction expansion. Let $ψ: \mathbb{N} \rightarrow \mathbb{N}$ be a function with $ψ(n)/n\to \infty$ as $n\to\infty$. The (upper, lower) fast Khintchine spectrum for $ψ$ is defined as the Hausdorff dimension of the set of numbers $x\in (0,1)$ for which the (upper, lower) limit of $\frac{1}{ψ(n)}\sum\_{j=1}^n\log a\_j(x)$ is equal to $1$. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.
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…
This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem 2.1, provides a rigorous estimate on the smallest value $t_1$ such that the portion of the Markov spectrum $(-\infty,t_1)\cap M$ has Hausdorff dimension $1$. Our second result, Theorem 3.1, gives a new upper bound on the Hausdorff dimension of the set difference $M\setminus L$. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets of continu…
Lüroth series, like regular continued fractions, provide an interesting identification of real numbers with infinite sequences of integers. These sequences give deep arithmetic and measure-theoretic properties of subsets of numbers according to their growth. Although different, regular continued fractions and Lüroth series share several properties. In this paper, we explore one similarity by estimating the Hausdorff dimension of subsets of real numbers whose Lüroth expansion grows at a definite rate. This is an extension of a result of Y. Sun and J. Wu to the context of Lüroth series. It was r…
We propose a new point of view on multidimensional continued fraction algorithms inspired by Rauzy induction. The generic behaviour of such an algorithm is described here as a random walk on a graph that we call simplicial system. These systems provide a family of examples for random walks with memory recorded by a finite dimensional vector. We introduce a general criterion on these graphs that induces ergodicity together with a bundle of many other dynamical properties. In particular, after computing the representation of Brun, Selmer and Arnoux-Rauzy-Poincare algorithm in this formalism, it …
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^{τ(x)(h(x)+\cdots+h(T^{n-1}(x)))} {align*} holds for infinitely many $n\in\mathbb{N},$ where $h$ and $τ$ are positive continuous functions, $T$ is the Gauss map and $a_n(x)$ denote the $n$th partial quotient of $x$ in its continued fraction expansion. By appropriate choices of $r$, $τ(x)$ snd $h(x)$ we obtain the classical Jarník-Besicovitch Theorem as well as more recent results by Wang-Wu-Xu, Wang-Wu, H…
In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if $E$ is any arithmetic progression, the set of primes, or the set of squares $\{n^2\}_{n \in \mathbb{N}}$, then the continued fractions whose digits lie in $E$ have full dimension spectrum, which we denote by $DS(\mathcal{CF}_E)$. Moreover we prove that if $E$ is an infinite set of consecutive powers then the dimension spectrum $DS(\mathcal{CF}_E)$ always contains a non trivial interval. We also show that there exists some $E \subset \mathbb{…
Hausdorff dimension for the weighted products of multiple digits in d-decaying Gauss like systems
本文研究了在d衰减高斯类迭代函数系统中多重数字加权乘积的Hausdorff维度，无需假设有界变形性质，解决了多重数字乘积的维度问题。
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 $n\to\infty$.
Fractal transference principle for continued fractions of Laurent series
本文基于Laurent级数连分数展开，建立分形转移原理，证明了在特定条件下，存在一个子集其连分数数字具有特定维度，并保持相对上密度。
Dynamics of Ostrowski skew-product: I. Limit laws and Hausdorff dimensions
本研究利用转移算子研究Ostrowski斜积动力系统，证明其存在具分段全纯密度的绝对连续不变测度与指数混合性，推导了算术随机变量的中心极限定理，并得到有界数字集Hausdorff维数的Bowen–Ruelle型谱隐式估计。
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 consecut…
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 $n\to\infty$.
Multifractal analysis of maximal product of consecutive partial quotients in continued fractions
本文从多重分形分析角度研究连分数前n项连续部分商最大乘积的增长速率，确定了对应水平集的Hausdorff维数，发现ρ=1/2时维数存在跳跃，并构造不连续函数填补了该维数间隙。
Metrical properties of finite product of partial quotients in arithmetic progressions
本文研究实数连分数中算术级数位置部分商乘积的度量性质，确定了满足乘积增长条件的实数集合的勒贝格测度与豪斯多夫维数，并揭示高斯动力系统多重遍历平均收敛定理例外集的大小。
Möbius inversion and coprime summation for error-sum functions of continued fractions
本文研究了连分数的无权误差和函数，利用Möbius反演和互素渐近分母求和等数论方法，证明其图像的Hausdorff维数恰好为1，并重新推导了相对误差和函数维数的上界。
We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Haus…
Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system
本文从多重分形角度研究 Schneider $p$-进连分数数字的增长速率，完全确定了其上极限归一化为 1 的集合 Hausdorff 维数，并推广到兼具下、上极限条件的交集集合。
This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$. Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$. By imposing $n_{_N}\le m \forall N\in \zN$, Jarnik defined the corresponding sets $E_m$ and gave a first estimate of $d_H(E_m)$, $d_H$ the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with $d_H(E_m)$ and $d_H(F_m)$, $F_m$ being the set of real numbers for which ${\sum_{i=1}^N n_i\over N}\le m$.
Uniform Diophantine approximation and run-length function in continued fractions
本文研究连分数中一致逼近指数与渐近逼近指数的多重分形性质，计算代数无理点对应的一致丢番图集合的豪斯多夫维数，并将结果应用于连分数游程函数的多重分形分析。
Metric Theory for Continued Fractions with Multiple Large Partial Quotients
本文研究连分数展开前n项中至少含有r个大偏商的实数集合，确定其Lebesgue测度和Hausdorff维数，解决了多个大偏商情形下经典极限定理失效的问题。
Metric properties of continued fractions with large prime partial quotients
研究连分数展开中存在至少两个同时大且素数的部分商的数集，建立其勒贝格测度的零一定律并确定其Hausdorff维度。
Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the improvements to Dirichlet's theorem (1842). We establish both the weak and strong laws of large numbers for the partial sums $S_n(x)= \sum_{i=1}^n a_i(x)a_{i+1}(x)$ as well as, from a multifractal analysis point of view, investigate its increasing rate. Specifically, we prove the following results: \medskip \begin{itemize} \item For any $ε>0$, the Lebesgue me…
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 …
Let $ω=[a_1, a_2, \cdots]$ be the infinite expansion of continued fraction for an irrational number $ω\in (0,1)$; let $R_n (ω)$ (resp. $R_{n, \, k} (ω)$, $R_{n, \, k+} (ω)$) be the number of distinct partial quotients each of which appears at least once (resp. exactly $k$ times, at least $k$ times) in the sequence $a_1, \cdots, a_n$. In this paper it is proved that for Lebesgue almost all $ω\in (0,1)$ and all $k \geq 1$, $$ \displaystyle \lim_{n \to \infty} \frac{R_n (ω)}{\sqrt{n}}=\sqrt{\fracπ{\log 2}}, \quad \lim_{n \to \infty} \frac{R_{n, \, k} (ω)}{R_n (ω)}=\frac{C_{2 k}^k}{(2k-1) \cdot 4^…
We initiate the study of the sets $H(c)$, $0<c<1$, of real $x$ for which the sequence $(kx)_{k\geq1}$ (viewed mod 1) consistently hits the interval $[0,c)$ at least as often as expected (i. e., with frequency $\geq c$). More formally, \[ H(c)=\{α\in \mathbf R\mid {\rm card}(\{1\leq k\leq n\mid < kα><c\})\geq cn, {for all}n\geq1\}. \] where $<x>=x-[x]$ stands for the fractional part of $x\in \mathbb R$. We prove that, for rational $c$, the sets $H(c)$ are of positive Hausdorff dimension and, in particular, are uncountable. For integers $m\geq1$, we obtain a surprising characterization of the nu…
A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the string of digits resulting from writing $a_i$ in base $b$. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set $T_b$ of Trott numbers is a complete $G_δ$ set. We prove moreover that the union $T:=\bigcup_{b\geq 2} T_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, w…
For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ be its continued fraction expansion. Denote by $T_n(x):= max \{a_k(x): 1\leq k\leq n\}$ the leading partial quotient up to $n$. For any real $α\in(0,\infty), γ\in(0,\infty)$, let $F(γ,α):=\{x\in(0,1)\setminus\mathbb{Q}: \lim_{n\rightarrow\infty}\frac{T_n(x)}{e^{n^γ}}=α\}$. For a set $E\subset (0,1)\setminus\mathbb{Q}$, let $dim_H E$ be its Hausdorff dimension. Recently Lingmin Liao and Michal Rams [LR, Theorem 1.3] show that $dim_H F(γ,α)$ is $1$ if $r\in(0,1/2)$, it is $1/2$ if $r\in(1/2,\infty)$ for any $α\in(0,\infty)…
Multifractal analysis of power means for the Schneider map on $p\mathbb{Z}_p$
研究p进数上Schneider继续分数映射系数的渐近幂均值，利用热力学正则化工具计算对应水平集的Hausdorff维度，并获得关联的多重分形谱的显式公式。
We look at the rate of growth of the partial quotients of the infinite continued fraction expansion of an irrational number relative to the rate of approximation of the number by its convergents. In non-generic cases the Hausdorff dimension of some exceptional sets is computed.
On the structure of the dimension spectrum for continued fraction expansions
本文研究了继续分数展开系数受限于无限子集时的维度谱结构，证明了某些集合的维度谱完整，并探讨了不同生成方式下的维度谱特性。
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 $ϕ(n)$ t…
Bounded continued fractions and quadratic orders in $\Q(\sqrt{127})$
本文研究Q(√127)中部分商≤4的有界连分数的周期与乘子序，证明相关精确公式，结合Hensley定理得到二次域允许无穷多此类连分数的数量下界。
On some Diophantine and Ergodic properties of the Schneider map over function fields
该论文研究函数域上Schneider映射定义的连分数，证明其Laurent级数域元素的连分数展开存在唯一，确立相关丢番图逼近与遍历性质，确定最难逼近元素集的Hausdorff维数，为函数域数论与动力系统提供新结果。
Sharp Hausdorff Dimension Bounds for Sets with Bounded and Growing Digits in $N$-expansions
本文研究了(0,1)区间内无理数集合的Hausdorff维数，针对数字有界或趋向无穷的情况，给出了精确的维数界，并改进了Jarník型结果，证明了数字趋向无穷的集合维数为1/2。
Metrical properties of Hurwitz Continued Fractions
本文研究Hurwitz连分数的几何与度量理论，证明有效序列空间非闭，并计算部分商绝对值对无穷多个$n$不低于任意给定函数$Φ(n)$的复数集合的Hausdorff维数。
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