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Kan Jiang

Publications and source records attributed to Kan Jiang.

34 records · Page 2Linked to original sources
arXiv2019

Visibility of Cartesian products of Cantor sets

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.

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arXiv2019

On arithmetic progressions in self-similar sets

Given a sequence $\{b_{i}\}_{i=1}^{n}$ and a ratio $λ\in (0,1),$ let $E=\cup_{i=1}^n(λE+b_i)$ be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in $E$. Our main idea is from the multiple $β$-expansions.

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arXiv2019

Multiple representations of real numbers on self-similar sets with overlaps

Let $K$ be the attractor of the following IFS $$\{f_1(x)=λx, f_2(x)=λx +c-λ,f_3(x)=λx +1-λ\}, $$ where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main results of this paper are as follows: $$\sqrt{K}+\sqrt{K}=[0,2]$$ if and only if $$\sqrt{c}+1\geq 2\sqrt{1-λ},$$ where $\sqrt{K}+\sqrt{K}=\{\sqrt{x}+\sqrt{y}:x,y\in K\}$. If $c\geq (1-λ)^2$, then $$\dfrac{K}{K}=\left\{\dfrac{x}{y}:x,y\in K, y\neq 0\right\}=\left[0,\infty\right).$$ As a consequence, we prove that the following conditions are equivalent: (1) For any $u\in [0,1]$, there are some $x,y\in K$ such that $u=x\cdot y;$ (2) For any $u\in [0,1]$, there are some $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8, x_9,x_{10}\in K$ such that $$u=x_1+x_2=x_3-x_4=x_5\cdot x_6=x_7÷x_8=\sqrt{x_9}+\sqrt{x_{10}};$$ (3) $c\geq (1-λ)^2$.

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arXiv2018

Interiors of continuous images of the middle-third Cantor set

Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the image \begin{equation*} f_{U}(C,C)=\{f(x,y):(x,y)\in (C\times C)\cap U\}. \end{equation*} If $\partial _{x}f$, $\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\in (C\times C)\cap U$ such that \begin{equation*} 1<\left\vert \frac{\partial _{x}f|_{(x_{0},y_{0})}}{\partial _{y}f|_{(x_{0},y_{0})}}\right\vert <3\text{ or }1<\left\vert \frac{\partial _{y}f|_{(x_{0},y_{0})}}{\partial _{x}f|_{(x_{0},y_{0})}}\right\vert <3, \end{equation*} then $f_{U}(C,C)$ has a non-empty interior. As a consequence, if \begin{equation*} f(x,y)=x^{α}y^{β}(αβ\neq 0),\text{ }x^{α}\pm y^{α}(α\neq 0)\text{ or }\sin (x)\cos (y), \end{equation*} then $f_{U}(C,C)$ contains a non-empty interior.

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arXiv2018

Arithmetic representations of real numbers in terms of self-similar sets

Suppose $n\geq 2$ and $\mathcal{A}_{i}\subset \{0,1,\cdots ,(n-1)\}$ for $ i=1,\cdots ,l,$ let $K_{i}=\bigcup\nolimits_{a\in \mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\cdots ,m_{l}\in \mathbb{Z}$ with $\prod\nolimits_{i}m_{i}\neq 0,$ we let \begin{equation*} S_{x}=\left\{ \mathbf{(}y_{1},\cdots ,y_{l}\mathbf{)}:m_{1}y_{1}+\cdots +m_{l}y_{l}=x\text{ with }y_{i}\in K_{i}\text{ }\forall i\right\} . \end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set \begin{equation*} U_{r}=\{x:\mathbf{Card}(S_{x})=r\}, \end{equation*} where $\mathbf{Card}(S_{x})$ denotes the cardinality of $S_{x}$, and $r\in \mathbb{N}^{+}$. We prove under the so-called covering condition that the Hausdorff dimension of $U_{1}$ can be calculated in terms of some matrix. Moreover, if $r\geq 2$, we also give some sufficient conditions such that the Hausdorff dimension of $U_{r}$ takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of $U_{r}$ is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.

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arXiv2018

Lipschitz equivalence of self-similar sets with exact overlaps

In this paper, we study a class $\mathcal{A}(λ,n,m)$ of self-similar sets with $m$ exact overlaps generated by $n$ similitudes of the same ratio $ λ$. We obtain a necessary condition for a self-similar set in $\mathcal{A}(λ,n,m)$ to be Lipschitz equivalent to a self-similar set satisfying the strong separation condition, i.e., there exists an integer $ k\geq 2$ such that $x^{2k}-mx^{k}+n$ is reducible, in particular, $m$ belongs to $\{a^{i}:a\in \mathbb{N}$ with $i\geq 2\}.$

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arXiv2018

Multiple expansions of real numbers with digits set $\{0,1,q\}$

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.

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arXiv2018

Multiplication on self-similar sets with overlaps

Let $A,B\subset\mathbb{R}$. Define $$A\cdot B=\{x\cdot y:x\in A, y\in B\}.$$ In this paper, we consider the following class of self-similar sets with overlaps. Let $K$ be the attractor of the IFS $\{f_1(x)=λx, f_2(x)=λx+c-λ,f_3(x)=λx+1-λ\}$, where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main result of this paper is $K\cdot K=[0,1]$ if and only if $(1-λ)^2\leq c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in[0,1]$, $u=x\cdot y$, where $x,y\in K$.

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arXiv2018

Projections of cartesian products of the self-similar sets without the irrationality assumption

Let $β>1$. Define a class of similitudes \[S=\left\{f_{i}(x)=\dfrac{x}{β^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Let $\mathcal{A}$ be the collection of all the self-similar sets generated by the similitudes from $S$. In this paper, we prove that for any $θ\in(0,π)$ and $K_1, K_2\in \mathcal{A}$, $Proj_θ(K_1\times K_2)$ is similar to a self-similar set or an attractor of some infinite iterated function system, where $Proj_θ$ denotes the orthogonal projection onto $L_θ$, and $L_θ$ denotes the line through the origin in direction $θ$. As a corollary, $\dim_{P}(Proj_θ(K_1\times K_2))=\overline{\dim}_{B}(Proj_θ(K_1\times K_2))$ holds for any $θ\in(0,π)$ and any $K_1, K_2\in \mathcal{A}$, where $\dim_{P}$ and $\overline{\dim}_{B}$ denote the packing and upper box dimension. Whether $Proj_θ(K_1\times K_2)$ is similar to a self-similar set or not is uniquely determined by the similarity ratios of $K_1$ and $K_2$ rather than the angle $θ.$ When $Proj_θ(K_1\times K_2)$ is similar to a self-similar set, in terms of the finite type condition \cite{NW}, we are able to calculate in cerntain cases the Hausdorff dimension of $Proj_θ(K_1\times K_2)$. If $Proj_θ(K_1\times K_2)$ is similar to an attractor of some infinite iterated function system, then by virtue of the Vitali covering lemma \cite{FG} we give an estimation of the Hausdorff dimension of $Proj_θ(K_1\times K_2)$. For some cases, we can calculate, by means of Mauldin and Urbanski' result \cite{MRD}, the exact Hausdorff dimension of $Proj_θ(K_1\times K_2)$. We also find some non-trivial examples such that for some angle $θ\in[0,π)$ and some $K_1, K_2\in \mathcal{A}$, $\dim_{H}(Proj_θ(K_1\times K_2))=\dim_{H}(K_1)+\dim_{H}(K_2)$.

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arXiv2017

On the points without universal expansions

Let $1<β<2$. Given any $x\in[0, (β-1)^{-1}]$, a sequence $(a_n)\in\{0,1\}^{\mathbb{N}}$ is called a $β$-expansion of $x$ if $x=\sum_{n=1}^{\infty}a_nβ^{-n}.$ For any $k\geq 1$ and any $(b_1b_2\cdots b_k)\in\{0,1\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1b_2\cdots b_k$, then we call $(a_n)$ a universal $β$-expansion of $x$. Sidorov \cite{Sidorov2003}, Dajani and de Vries \cite{DajaniDeVrie} proved that given any $1<β<2$, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the set $V_β$ of points without universal expansions. For any $n\geq 2$, let $β_n$ be the $n$-bonacci number satisfying the following equation: $β^n=β^{n-1}+β^{n-2}+\cdots +β+1.$ Then we have $\dim_{H}(V_{β_n})=1$, where $\dim_{H}$ denotes the Hausdorff dimension. Similar results are still available for some other algebraic numbers. As a corollary, we give some results of the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper \cite{KarmaKan}.

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arXiv2016

Subshift of finite type and self-similar sets

Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that $K$ can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of $K$ as well as the set of elements in $K$ with unique codings using the machinery of Mauldin and Williams \cite{MW}. We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of $K$ with multiple codings. Secondly, in the setting of $β$-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. Motivated by this application, we prove that the set of all the unique codings is a subshift of finite type if and only if it is a sofic shift. This equivalent condition was not mentioned by de Vries and Komornik \cite[Theorem 1.8]{MK}. Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the survivor set can be identified with a subshift of finite type. The third application partially answers a problem posed by Alcaraz Barrera \cite{Barrera}.

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arXiv2016

Shrinking random $β$-transformation

For any $n\geq 3$, let $1<β<2$ be the largest positive real number satisfying the equation $$β^n=β^{n-2}+β^{n-3}+\cdots+β+1.$$ In this paper we define the shrinking random $β$-transformation $K$ and investigate natural invariant measures for $K$, and the induced tranformation of $K$ on a special subset of the domain. We prove that both transformations have a unique measure of maximal entropy. However, the measure induced from the intrinsically ergodic measure for $K$ is not the intrinsically ergodic measure for the induced system.

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arXiv2016

Hausdorff dimension of the arithmetic sum of self-similar sets

Let $β>1$. We define a class of similitudes \[S:=\left\{f_{i}(x)=\dfrac{x}{β^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Taking any finite similitudes $\{f_{i}(x)\}_{i=1}^{m} $ from $S$, it is well known that there is a unique self-similar set $K_1$ satisfying $K_1=\cup_{i=1}^{m} f_{i}(K_1)$. Similarly, another self-similar set $K_2$ can be generated via the finite contractive maps of $S$. We call $K_1+K_2=\{x+y:x\in K_1, y\in K_2\}$ the arithmetic sum of two self-similar sets. In this paper, we prove that $K_1+K_2$ is either a self-similar set or a unique attractor of some infinite iterated function system. Using this result we can then calculate the exact Hausdorff dimension of $K_1+K_2$ under some conditions, which partially provides the dimensional result of $K_1+K_2$ if the IFS's of $K_1$ and $K_2$ fail the irrational assumption, see Peres and Shmerkin \cite{PS}.

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arXiv2014

On univoque points for self-similar sets

Let $K\subseteq\mathbb{R}$ be the unique attractor of an iterated function system. We consider the case where $K$ is an interval and study those elements of $K$ with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence of this, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams \cite{MW}. The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of \cite{DKK}, \cite{K1}, \cite{K2}, and \cite{MK}.

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