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Self-similar subsets of the Cantor set

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De-Jun Feng, Hui Rao, Yang Wang

Dynamical Systems · Submitted 20 Jun 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$.

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arXiv ID1406.5314 (version 1)
Pith IDB6DA56MI, Pith's permanent record of this paper and its review
CitationDe-Jun Feng, Hui Rao, Yang Wang. "Self-similar subsets of the Cantor set." arXiv:1406.5314 (2014). https://pith.science/paper/B6DA56MI
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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.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,625 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.71 · compares with prior results 0.12 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$. In particular, a simple criterion is formulated to characterize self-similar subsets of $\C$ with equal contraction ratio in modulus.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? background 0.91 0.33 0.04 unclear 0.52
s1 We give criteria for a complete classification of all such subsets. contribution 0.96 0.94 0.24 asserted 1.00
s2 We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$. result 1.00 0.94 0.60 asserted 1.00
s3 In particular, a simple criterion is formulated to characterize self-similar subsets of $\C$ with equal contraction ratio in modulus. result 0.74 0.93 0.15 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • references and citing sentences
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  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$. In particular, a simple criterion is formulated to characterize self-similar subsets of $\C$ with equal contraction ratio in modulus.

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