Kan Jiang

activity

20142022

most citedMultiple expansions of real numbers with digits set

{0, 1, q}

7 citations · 55 across the 25 of their papers we have counts for

collaborators

27 papers

math.DS2022★ 1 cited

Thickness theorems with partial derivatives

Kan Jiang

In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor…

math.NT2022★ 4 cited

Rational Points in Translations of The Cantor Set

Kan Jiang, Derong Kong, Wenxia Li +1

Given two coprime integers $p \geq 2$ and $q \geq 3$ , let $D_{p} \subset [0, 1)$ consist of all rational numbers which have a finite $p$ -ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{…

math.DS2021

On a class of self-similar sets which contain finitely many common points

Kan Jiang, Derong Kong, Wenxia Li +1

For $λ \in (0, 1/2]$ let $K_{λ} \subset R$ be a self-similar set generated by the iterated function system ${λ x, λ x + 1 - λ}$ . Given $x \in (0, 1/2)$ , let $Λ (x)$ be the set of $λ\in(0…

math.DS2021★ 5 cited

How likely can a point be in different Cantor sets

Kan Jiang, Derong Kong, Wenxia Li

Let $m \in N_{\geq 2}$ , and let $K = {K_{λ} : λ \in (0, 1/ m]}$ be a class of Cantor sets, where $K_{λ} = {\sum_{i = 1}^{\infty} d_{i} λ^{i} : d_{i} \in {0, 1, \dots, m - 1}, i \geq 1}$ . We i…

math.NT2021★ 1 cited

On some fractal sets only containing irrationals

Kan Jiang

In this paper, we prove that many fractal sets generated by the associated dynamical systems only contain irrationals. As an application, we explicitly construct some overlapping s…

math.NT2020

Irrational self-similar sets

Qi Jia, Yuanyuan Li, Kan Jiang

Let $K \subset R$ be a self-similar set defined on $R$ . It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$ , $$K+t=\…

math.DS2020

A nonlinear version of the Newhouse thickness theorem

Kan Jiang

Let $C_{1}$ and $C_{2}$ be two Cantor sets with convex hull $[0, 1]$ . Newhouse proved if $τ (C_{1}) \cdot τ (C_{2}) \geq 1$ , then the arithmetic sum $C_{1} + C_{2}$ is an interval, where $τ(C_i), 1\l…

math.MG2020

On continuous images of self-similar sets

Yuanyuan Li, Jiaqi Fan, Jiangwen Gu +2

Let $(M, c_{k}, n_{k}, κ)$ be a class of homogeneous Moran sets. Suppose $f (x, y) \in C^{3}$ is a function defined on $R^{2}$ . Given $E_{1}, E_{2} \in (M, c_{k}, n_{k}, κ)$ …

math.DS2020★ 3 cited

On the sum of squares of middle-third Cantor set

Zhiqiang Wang, Kan Jiang, Wenxia Li +1

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 a…

math.DS2019★ 5 cited

Multiplication on uniform $λ$ -Cantor sets

Jiangwen Gu, Kan Jiang, Lifeng Xi +1

Let $C$ be the middle-third Cantor set. Define $C * C = {x * y : x, y \in C}$ , where $* = +, -, \cdot, \div$ (when $* = \div$ , we assume $y \neq = 0$ ). Steinhaus \cite{HS} proved in 1917 that \[ C-C=[-1,1]…

math.DS2019

Arithmetic on self-similar sets

Bing Zhao, Xiaomin Ren, Jiali Zhu +1

Let $K_{1}$ and $K_{2}$ be two one-dimensional homogeneous self-similar sets. Let $f$ be a continuous function defined on an open set $U \subset R^{2}$ . Denote the continuous i…

math.DS2019★ 3 cited

Visibility of Cartesian products of Cantor sets

Tingyu Zhang, Kan Jiang, Wenxia Li

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 $y = α x$ is visible thro…

math.DS2019★ 3 cited

Arithmetic on Moran sets

Xiaomin Ren, Li Tian, Jiali Zhu +1

Let $(M, c_{k}, n_{k})$ be a class of Moran sets. We assume that the convex hull of any $E \in (M, c_{k}, n_{k})$ is $[0, 1]$ . Let $A, B$ be two non-empty sets in $\mathbb{R…

math.NT2019

On arithmetic progressions in self-similar sets

Kan Jiang, Qiyang Pei, Lifeng Xi

Given a sequence ${b_{i}}_{i = 1}^{n}$ and a ratio $λ \in (0, 1),$ let $E = \cup_{i = 1}^{n} (λ E + b_{i})$ be a homogeneous self-similar set. In this paper, we study the existence and maximal l…

math.DS2018

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

Xiaomin Ren, Jiali Zhu, Li Tian +1

Let $K$ be the attractor of the following IFS ${f_{1} (x) = λ x, f_{2} (x) = λ x + c - λ, f_{3} (x) = λ x + 1 - λ},$ where $f_{1} (I) \cap f_{2} (I) \neq = \emptyset, (f_{1} (I) \cup f_{2} (I)) \cap f_{3} (I) = \emptyset,$ …

math.DS2018★ 6 cited

Interiors of continuous images of the middle-third Cantor set

Kan Jiang, Lifeng Xi

Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U \subset R^{2}$ . Denote the image \begin{equation*} f_{U}(C,C)=\{f(x,y):(x,y)…

math.DS2018★ 2 cited

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

Kan Jiang, Lifeng Xi

Suppose $n \geq 2$ and $A_{i} \subset {0, 1, \dots, (n - 1)}$ for $i = 1, \dots, l,$ let $K_{i} = ⋃_{a \in A_{i}} n^{- 1} (K_{i} + a)$ be self-similar sets c…

math.DS2018★ 6 cited

Lipschitz equivalence of self-similar sets with exact overlaps

Kan Jiang, Songjing Wang, Lifeng Xi

In this paper, we study a class $A (λ, n, m)$ of self-similar sets with $m$ exact overlaps generated by $n$ similitudes of the same ratio $λ$ . We obtain a necessary conditi…

math.DS2018★ 2 cited

Multiplication on self-similar sets with overlaps

Li Tian, Jiangwen Gu, Qianqian Ye +2

Let $A, B \subset R$ . 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 attra…

math.DS2018★ 4 cited

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

Kan Jiang

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 $A$ be the collection of all t…

math.DS2017

Estimating the Hausdorff dimensions of univoque sets for self-similar sets

Xiu Chen, Kan Jiang, Wenxia Li

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 univoqu…

math.DS2017

On the points without universal expansions

Karma Dajani, Kan Jiang

Let $1 < β < 2$ . Given any $x \in [0, (β - 1)^{- 1}]$ , a sequence $(a_{n}) \in {0, 1}^{N}$ is called a $β$ -expansion of $x$ if $x = \sum_{n = 1}^{\infty} a_{n} β^{- n} .$ For any $k \geq 1$ and…

math.DS2016

Shrinking random $β$ -transformation

Kan Jiang, Karma Dajani

For any $n \geq 3$ , let $1 < β < 2$ be the largest positive real number satisfying the equation $β^{n} = β^{n - 2} + β^{n - 3} + \dots + β + 1.$ In this paper we define the shrinking random $β$ -tran…

math.DS2015★ 3 cited

Subshift of finite type and self-similar sets

Kan Jiang, Karma Dajani

Let $K \subset R$ 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 subshi…

Thickness theorems with partial derivatives

Rational Points in Translations of The Cantor Set

On a class of self-similar sets which contain finitely many common points

How likely can a point be in different Cantor sets

On some fractal sets only containing irrationals

Irrational self-similar sets

A nonlinear version of the Newhouse thickness theorem

On continuous images of self-similar sets

On the sum of squares of middle-third Cantor set

Multiplication on uniform λ-Cantor sets

Arithmetic on self-similar sets

Visibility of Cartesian products of Cantor sets

Arithmetic on Moran sets

On arithmetic progressions in self-similar sets

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

Interiors of continuous images of the middle-third Cantor set

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

Lipschitz equivalence of self-similar sets with exact overlaps

Multiplication on self-similar sets with overlaps

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

Estimating the Hausdorff dimensions of univoque sets for self-similar sets

On the points without universal expansions

Shrinking random β-transformation

Subshift of finite type and self-similar sets

Multiplication on uniform $λ$ -Cantor sets

Shrinking random $β$ -transformation