RECORD
Multiple codings for self-similar sets with overlaps
Dynamical Systems · Submitted 30 Mar 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(\Phi)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(\Phi)$ is always infinite for any $k\ge 2$.
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
In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $\Phi=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ 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 $\mathcal U_k(\Phi)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(\Phi)$ is closed if and only if $\mathcal U_{\aleph_0}(\Phi)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(\Phi)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(\Phi)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(\Phi)$ and ${U_{\aleph_0}(\Phi)}$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. | background 0.60 | 0.52 | 0.02 | asserted 0.87 |
| s1 | Given a self-similar iterated function system $\Phi=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ 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 $$ x=\lim_{n\to\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). | background 0.89 | 0.19 | 0.03 | asserted 0.96 |
| s2 | $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(\Phi)$ which consists of all $x\in E$ having precisely $k$ different codings. | contribution 0.96 | 0.77 | 0.07 | asserted 0.83 |
| s3 | Among several equivalent characterizations we show that $\mathcal U_1(\Phi)$ is closed if and only if $\mathcal U_{\aleph_0}(\Phi)$ is an empty set. | result 0.99 | 0.95 | 0.21 | asserted 1.00 |
| s4 | Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(\Phi)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(\Phi)$ is always infinite for any $k\ge 2$. | result 0.98 | 0.96 | 0.27 | asserted 1.00 |
| s5 | Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(\Phi)$ and ${U_{\aleph_0}(\Phi)}$. | result 0.88 | 0.94 | 0.15 | 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
In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $\Phi=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ 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 $\mathcal U_k(\Phi)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(\Phi)$ is closed if and only if $\mathcal U_{\aleph_0}(\Phi)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(\Phi)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(\Phi)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(\Phi)$ and ${U_{\aleph_0}(\Phi)}$.
Human review
How the literature uses it
10 arXiv papers cite this in 14 sentences
Papers citing it · 7 sentences without a confident label
Uses its method or tool 1 paper
-
The proof is similar to that of [ref] .
On a kind of self-similar sets with complete overlaps · Spectrum and unique/multiple expansions · 0.84
Cites it as background 5 papers
-
The deeper results can be found in [ref] .
Subshift of finite type and self-similar sets · Points in $K$ with multiple codings · 0.94
-
We note that finding the arithmetic progressions in $E_λ^n$ is essentially a problem in the setting of $β$-expansions, i.e. given a point in some interval, then how can we find its expansions in base $1/λ.$ Nevertheless, for the multiple $β$-expansions, to the best of our knowledge, there are few results [ref] .
On arithmetic progressions in self-similar sets · Introduction · 0.80
-
In [ref] , Dajani et al. analyzed the points in $Λ$ with multiple codings, and obtained that when the overlaps are the exact overlaps, then the set of points with exactly $k$ codings has the same Hausdorff dimension as the univoque set.
Multiplication on self-similar sets with overlaps · Introduction · 0.96
What it cites · 4 sentences without a confident label
Builds on 1 paper
-
In this paper we generalize [ref] and obtain the following results.
Multiple expansions of real numbers with digits set $\{0,1,q\}$ · Introduction · 1.00
Uses the method or tool of 1 paper
-
Therefore we can define an adjacency matrix for the univoque set, see [ref] , i.e. [equation] Using these two matrices, we can explicitly calculate the Hausdorff dimension of $E$ as well as $U_1(E)$, see [ref] .
Subshift of finite type and self-similar sets · Uncountable codings · 0.93
Cites as background 2 papers
-
Later, Dajani and Jiang [ref] considered the calculation of the Hausdorff dimension of $_1(E)$.
Subshift of finite type and self-similar sets · Introduction · 0.86
-
In particular, for $k=1$ there are many works devoted to $_1$ (cf. [ref] ).
Hausdorff dimension of univoque sets and Devil's staircase · Introduction · 0.95
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.