RECORD
On the union of homogeneous symmetric Cantor set with its translations
Dynamical Systems · Submitted 4 Feb 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Furthermore, for $0< \beta < 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Fix a positive integer $N$ and a real number $0< \beta < 1/(N+1)$. Let $\Gamma$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big\{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\cdots, N \Big\}. $$ For $m\in\mathbb{Z}_+$ we show that there exist infinitely many translation vectors $\mathbf t=(t_0,t_1,\cdots, t_m)$ with $0=t_0<t_1<\cdots<t_m$ such that the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. Furthermore, for $0< \beta < 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Fix a positive integer $N$ and a real number $0< \beta < 1/(N+1)$. | background 0.96 | 0.17 | 0.02 | unclear 0.65 |
| s1 | Let $\Gamma$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big\{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\cdots, N \Big\}. | background 0.81 | 0.34 | 0.02 | asserted 0.65 |
| s2 | $$ For $m\in\mathbb{Z}_+$ we show that there exist infinitely many translation vectors $\mathbf t=(t_0,t_1,\cdots, t_m)$ with $0=t_0<t_1<\cdots<t_m$ such that the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. | result 0.95 | 0.96 | 0.68 | asserted 1.00 |
| s3 | Furthermore, for $0< \beta < 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. | result 0.82 | 0.95 | 0.42 | asserted 1.00 |
| s4 | Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent. | method 0.99 | 0.91 | 0.10 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Fix a positive integer $N$ and a real number $0< \beta < 1/(N+1)$. Let $\Gamma$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big\{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\cdots, N \Big\}. $$ For $m\in\mathbb{Z}_+$ we show that there exist infinitely many translation vectors $\mathbf t=(t_0,t_1,\cdots, t_m)$ with $0=t_0<t_1<\cdots<t_m$ such that the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. Furthermore, for $0< \beta < 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.