arXiv subjects
Publications and source records attributed to Wenxia Li.
Let $C$ be the middle-third Cantor set. In this paper, we show that for every $x\in [0,4]$, there exist $x_1, x_2, x_3, x_4 \in C$ such that $$x= x_1^2+x_2^2+x_3^2+x_4^2,$$ which answers a question posed by Athreya, Reznick,and Tyson.
In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $Φ=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1i_2\ldots\in\{1,\ldots,m\}^\mathbb N$, called a coding of $x$, such that $$ x=\lim_{n\to\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(Φ)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(Φ)$ is closed if and only if $\mathcal U_{\aleph_0}(Φ)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(Φ)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(Φ)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(Φ)$ and ${U_{\aleph_0}(Φ)}$.
Let $K_λ$ be the attractor of the following IFS \begin{equation*} \{f_1(x)=λx, f_2(x)=λx+1-λ\}, \;\;0<λ<1/2. \end{equation*} Given $α\geq 0$, we say the line $y=αx$ is visible through $K_λ\times K_λ$ if $$ \{(x, αx): x\in \mathbb R\setminus \{0\}\}\cap ((K_λ\times K_λ))=\emptyset. $$ Let $V=\left \{α\geq 0: y=αx \mbox{ is visible through } K_λ\times K_λ \right \}$. In this paper, we give a completed description of $V$, e.g., its Hausdoff dimension and its topological property. Moreover, we also discuss another type of visible problem which is related to the slicing problem.
Given $β\in(1,2)$ the fat Sierpinski gasket $\mathcal S_β$ is the self-similar set in $\mathbb R^2$ generated by the iterated function system (IFS) \[ f_{β,d}(x)=\frac{x+d}β,\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\}. \] Then for each point $P\in\mathcal S_β$ there exists a sequence $(d_i)\in\mathcal A^\mathbb N$ such that $P=\sum_{i=1}^\infty d_i/β^i$, and the infinite sequence $(d_i)$ is called a \emph{coding} of $P$. In general, a point in $\mathcal S_β$ may have multiple codings since the overlap region $\mathcal O_β:=\bigcup_{c,d\in\mathcal A, c\ne d}f_{β,c}(Δ_β)\cap f_{β,d}(Δ_β)$ has non-empty interior, where $Δ_β$ is the convex hull of $\mathcal S_β$. In this paper we are interested in the invariant set \[ \widetilde{\mathcal U}_β:=\left\{\sum_{i=1}^\infty \frac{d_i}{β^i}\in \mathcal S_β: \sum_{i=1}^\infty\frac{d_{n+i}}{β^i}\notin\mathcal O_β~\forall n\ge 0\right\}. \] Then each point in $ \widetilde{\mathcal U}_β$ has a unique coding. We show that there is a transcendental number $β_c\approx 1.55263$ related to the Thue-Morse sequence, such that $\widetilde{\mathcal U}_β$ has positive Hausdorff dimension if and only if $β>β_{c}$. Furthermore, for $β=β_c$ the set $\widetilde{\mathcal U}_β$ is uncountable but has zero Hausdorff dimension, and for $β<β_c$ the set $\widetilde{\mathcal U}_β$ is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of $\widetilde{\mathcal U}_β$.
For $q>1$ we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let $\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\cdots,\aleph_0$ let $\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\in \mathcal{B}_k$ let $\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that \[ \mathcal{B}_{\aleph_0}=[2,\infty),\quad \mathcal{B}_k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2, \] where $q_c\approx 2.32472$ is the appropriate root of $x^3-3x^2+2x-1=0$. Moreover, we show that for any positive integer $k\ge 2$ and any $q\in\mathcal{B}_{k}$ the Hausdorff dimensions of $\mathcal{U}_q^{(k)}$ and $\mathcal{U}_q$ are the same, i.e., \[ \dim_H\mathcal{U}_q^{(k)}=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad k\ge 2. \] Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension.
Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding $q$-expansions. Recently, Komornik et al.~(Adv. Math., 2017) showed that the topological entropy function $H: q \mapsto h_{top}(\mathbf{U}_q)$ is a Devil's staircase in $(1, M+1]$. Let $\mathcal{B}$ be the bifurcation set of $H$ defined by \[ \mathcal{B}=\{q\in(1, M+1]: H(p)\ne H(q)\quad\textrm{for any}\quad p\ne q\}. \] In this paper we analyze the fractal properties of $\mathcal{B}$, and show that for any $q\in \mathcal{B}$, \[ \lim_{δ\rightarrow 0} \dim_H(\mathcal{B}\cap(q-δ, q+δ))=\dim_H\mathcal{U}_q, \] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when $q\in\mathcal{B}$ the univoque set $\mathcal{U}_q$ is dimensionally homogeneous, i.e., $ \dim_H(\mathcal{U}_q\cap V)=\dim_H\mathcal{U}_q $ for any open set $V$ that intersect $\mathcal{U}_q$. As an application we obtain a dimensional spectrum result for the set $\mathcal{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique $q$-expansion. In particular, we prove that for any $t>1$ we have \[ \dim_H(\mathcal{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also consider the variations of the sets $\mathcal{U}=\mathcal{U}(M)$ when $M$ changes.
Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. Lü, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+λ\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^λ$ contain an interval for all $x\in(0, 1]$ and $λ\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017).
For $β\in(1,2]$ the $β$-transformation $T_β: [0,1) \to [0,1)$ is defined by $T_β( x) = βx \pmod 1$. For $t\in[0, 1)$ let $K_β(t)$ be the survivor set of $T_β$ with hole $(0,t)$ given by \[K_β(t):=\{x\in[0, 1): T_β^n(x)\not \in (0, t) \textrm{ for all }n\ge 0\}.\] In this paper we characterise the bifurcation set $E_β$ of all parameters $t\in[0,1)$ for which the set valued function $t\mapsto K_β(t)$ is not locally constant. We show that $E_β$ is a Lebesgue null set of full Hausdorff dimension for all $β\in(1,2)$. We prove that for Lebesgue almost every $β\in(1,2)$ the bifurcation set $E_β$ contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of $β\in(1,2)$ for which $E_β$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_2$, the bifurcation set of the doubling map. Finally, we give for each $β\in (1,2)$ a lower and upper bound for the value $τ_β$, such that the Hausdorff dimension of $K_β(t)$ is positive if and only if $t< τ_β$. We show that $τ_β\le 1-\frac1β$ for all $β\in (1,2)$.
An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.
Given a positive integer $M$ and a real number $q >1$, a \emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $(c_i) \in \{0,\ldots,M\}^\infty$ such that \[x=\sum_{i=1}^{\infty} c_iq^{-i}.\] It is well known that if $q \in (1,M+1]$, then each $x \in I_q:=\left[0,M/(q-1)\right]$ has a $q$-expansion. Let $\mathcal{U}=\mathcal{U}(M)$ be the set of \emph{univoque bases} $q>1$ for which $1$ has a unique $q$-expansion. The main object of this paper is to provide new characterizations of $\mathcal{U}$ and to show that the Hausdorff dimension of the set of numbers $x \in I_q$ with a unique $q$-expansion changes the most if $q$ "crosses" a univoque base. Denote by $\mathcal{B}_2=\mathcal{B}_2(M)$ the set of $q \in (1,M+1]$ such that there exist numbers having precisely two distinct $q$-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (2009) and prove that \[\dim_H(\mathcal{B}_2\cap(q',q'+δ))>0\quad\textrm{for any}\quad δ>0,\] where $q'=q'(M)$ is the Komornik-Loreti constant.
We consider a class of homogeneous self-similar sets with complete overlaps and give a sufficient condition for the Lipschitz equivalence between members in this class.
Given two positive integers $M$ and $k$, let $\B_k$ be the set of bases $q>1$ such that there exists a real number $x$ having exactly $k$ different $q$-expansions over the alphabet $\{0,1,\cdots,M\}$. In this paper we investigate the smallest base $q_2$ of $\B_2$, and show that if $M=2m$ the smallest base $$q_2 =\frac{m+1+\sqrt{m^2+2m+5}}{2},$$ and if $M=2m-1$ the smallest base $q_2$ is the appropriate root of $$ x^4=(m-1)\,x^3+2 m\, x^2+m \,x+1. $$ Moreover, for $M=2$ we show that $q_2$ is also the smallest base of $\B_k$ for all $k\ge 3$. This turns out to be different from that for $M=1$.
We fix a positive integer $M$, and we consider expansions in arbitrary real bases $q>1$ over the alphabet $\{0,1,...,M\}$. We denote by $U_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension $D(q)$ of $U_q$ for each $q\in (1,\infty)$. Furthermore, we prove that the dimension function $D:(1,\infty)\to[0,1]$ is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in $(q',\infty)$, where $q'$ denotes the Komornik--Loreti constant: although $D(q)>D(q')$ for all $q>q'$, we have $D'<0$ a.e. in $(q',\infty)$. During the proofs we improve and generalize a theorem of Erdős et al. on the existence of large blocks of zeros in $β$-expansions, and we determine for all $M$ the Lebesgue measure and the Hausdorff dimension of the set of bases in which $x=1$ has a unique expansion.
Given an integer $N\ge 2$ and a real number $β>1$, let $Γ_{β,N}$ be the set of all $x=\sum_{i=1}^\infty {d_i}/{β^i}$ with $d_i\in\{0,1,\cdots,N-1\}$ for all $i\ge 1$. The infinite sequence $(d_i)$ is called a $β$-expansion of $x$. Let $\mathbf{U}_{β,N}$ be the set of all $x$'s in $Γ_{β,N}$ which have unique $β$-expansions. We give explicit formula of the Hausdorff dimension of $\mathbf{U}_{β,N}$ for $β$ in any admissible interval $[{β}_L,{β}_U]$, where ${β_L}$ is a purely Parry number while ${β_U}$ is a transcendental number whose quasi-greedy expansion of $1$ is related to the classical Thue-Morse sequence. This allows us to calculate the Hausdorff dimension of $\U{N}$ for almost every $β>1$. In particular, this improves the main results of G{á}bor Kall{ó}s (1999, 2001). Moreover, we find that the dimension function $f(β)=\dim_H\mathbf{U}_{β,N}$ fluctuates frequently for $β\in(1,N)$.
Let $Γ_{β,N}$ be the $N$-part homogeneous Cantor set with $β\in(1/(2N-1),1/N)$. Any string $(j_\ell)_{\ell=1}^\N$ with $j_\ell\in\{0,\pm 1,...,\pm(N-1)\}$ such that $t=\sum_{\ell=1}^\N j_\ellβ^{\ell-1}(1-β)/(N-1)$ is called a code of $t$. Let $\mathcal{U}_{β,\pm N}$ be the set of $t\in[-1,1]$ having a unique code, and let $\mathcal{S}_{β,\pm N}$ be the set of $t\in\mathcal{U}_{β,\pm N}$ which make the intersection $Γ_{β,N}\cap(Γ_{β,N}+t)$ a self-similar set. We characterize the set $\mathcal{U}_{β,\pm N}$ in a geometrical and algebraical way, and give a sufficient and necessary condition for $t\in\mathcal{S}_{β,\pm N}$. Using techniques from beta-expansions, we show that there is a critical point $β_c\in(1/(2N-1),1/N)$, which is a transcendental number, such that $\mathcal{U}_{β,\pm N}$ has positive Hausdorff dimension if $β\in(1/(2N-1),β_c)$, and contains countably infinite many elements if $β\in(β_c,1/N)$. Moreover, there exists a second critical point $α_c=\big[N+1-\sqrt{(N-1)(N+3)}\,\big]/2\in(1/(2N-1),β_c)$ such that $\mathcal{S}_{β,\pm N}$ has positive Hausdorff dimension if $β\in(1/(2N-1),α_c)$, and contains countably infinite many elements if $β\in[α_c,1/N)$.