Multiplication on uniform -Cantor sets
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\neq0 $). Steinhaus\cite {HS} proved in 1917 …
,\cdot,\div $(when $*=\div $, we assume $ y\neq0 $). Steinhaus\cite {HS} proved in 1917 …
Arithmetic representations of real numbers in terms of self-similar sets
K Jiang, L Xi - arXiv preprint arXiv:1808.09724, 2018 - arxiv.org
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 …
let $ K_ {i}=\bigcup\nolimits_ {a\in\mathcal {A} _ {i}} n^{-1}(K_ {i}+ a) $ be self-similar sets …
Interiors of continuous images of the middle-third Cantor set
K Jiang, L Xi - arXiv preprint arXiv:1809.01880, 2018 - arxiv.org
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 …
set $ U\subset\mathbb {R}^{2} $. Denote the image\begin {equation*} f_ {U}(C, C)=\{f (x, y):(x …
On continuous images of self-similar sets
Let (M, ck, nk, κ) be a class of homogeneous Moran sets. Suppose f (x, y)∈ C 3 is a function
defined on R 2. Given E 1, E 2∈(M, ck, nk, κ), in this paper, we prove, under some …
defined on R 2. Given E 1, E 2∈(M, ck, nk, κ), in this paper, we prove, under some …
Arithmetic on Moran sets
X Ren, L Tian, J Zhu, K Jiang - Fractals, 2019 - World Scientific
Let (ℳ, ck, nk) be a class of Moran sets. We assume that the convex hull of any E∈(ℳ, ck, nk)
is [0, 1]. Let A, B be two nonempty sets in ℝ. Suppose that f is a continuous function defined …
is [0, 1]. Let A, B be two nonempty sets in ℝ. Suppose that f is a continuous function defined …
Multiple representations of real numbers on self-similar sets with overlaps
X Ren, J Zhu, L Tian, K Jiang - arXiv preprint arXiv:1810.04930, 2018 - arxiv.org
Let $ K $ be the attractor of the following IFS $$\{f_1 (x)=\lambda x, f_2 (x)=\lambda x+ c-
\lambda, f_3 (x)=\lambda x+ 1-\lambda\}, $$ where $ f_1 (I)\cap f_2 (I)\neq\emptyset,(f_1 …
\lambda, f_3 (x)=\lambda x+ 1-\lambda\}, $$ where $ f_1 (I)\cap f_2 (I)\neq\emptyset,(f_1 …
Arithmetic Product Of Self-Similar Sets With Two Branches
Q Wang, K Zhang - Fractals, 2021 - World Scientific
For self-similar set E= λ 1 E∪(λ 2 E+ 1− λ 2)(λ 1+ λ 2< 1), we study the arithmetic product of
E and obtain that when λ 1≥ λ 2> 0 then E⋅ E=[0, 1] if and only if λ 1≥(1− λ 2) 2. To deal …
E and obtain that when λ 1≥ λ 2> 0 then E⋅ E=[0, 1] if and only if λ 1≥(1− λ 2) 2. To deal …
Visibility of Cartesian products of Cantor sets
T Zhang, K Jiang, W Li - arXiv preprint arXiv:1905.04811, 2019 - arxiv.org
Let $ K_ {\lambda} $ be the attractor of the following IFS\begin {equation*}\{f_1 (x)=\lambda
x, f_2 (x)=\lambda x+ 1-\lambda\},\;\; 0<\lambda< 1/2.\end {equation*} Given $\alpha\geq 0 …
x, f_2 (x)=\lambda x+ 1-\lambda\},\;\; 0<\lambda< 1/2.\end {equation*} Given $\alpha\geq 0 …