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**On some properties of the Cantor set.**
*(English)*
Zbl 0879.26003

The paper contains two results. The first one is the extension of an old result of V. Jarník (1936) saying that no irrational point of a Cantor set \(C\) can be represented as a midpoint of two distinct points of \(C\). The authors show that no non-endpoint of \(C\) can be a midpoint of two distinct Cantor points. (The non-endpoint is every point of \(C\) which is not endpoint of any of the remaining closed intervals in the construction of \(C\).)

The other result reads as follows: Given two positive numbers \(p\), \(q\), such that \(0<p/q<1\) and an interior point \(R\) of the square \(\langle 0,1 \rangle^2\) we can find a rectangle \(ABCD\) with vertices in \(C\times C\) such that \(R\) lies on \(AC\) dividing it in the ratio \(p:q\). The rectangle has automatically a certain additional property concerning the so-called cross ratio of the four concurrent straight lines.

The other result reads as follows: Given two positive numbers \(p\), \(q\), such that \(0<p/q<1\) and an interior point \(R\) of the square \(\langle 0,1 \rangle^2\) we can find a rectangle \(ABCD\) with vertices in \(C\times C\) such that \(R\) lies on \(AC\) dividing it in the ratio \(p:q\). The rectangle has automatically a certain additional property concerning the so-called cross ratio of the four concurrent straight lines.

Reviewer: Jaroslav Tišer (Praha)

### MSC:

26A03 | Foundations: limits and generalizations, elementary topology of the line |

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |

### References:

[1] | N.C. Bose Majumdar: Some new results on the distance set of the Cantor set. Bull. Cal. Math. Soc. 52 (1960), no. 1, 1-13. · Zbl 0100.04801 |

[2] | N.C. Bose Majumdar: On the distance set of the Cantor middle third set - III. Amer. Math. Monthly 72 (1965), 725. · Zbl 0154.05403 · doi:10.2307/2314413 |

[3] | D.K. Ganguly: Generalization of some known properties of Cantor set. Czechoslovak Math. Jour. 28 (1978), no. 103, 369-372. · Zbl 0408.04001 |

[4] | V. Jarník: Sur les fonctions de variables reeles. Fund. Math. 27 (1936), 147-150. · Zbl 0015.10403 |

[5] | J. Randolph: Distance between points of the Cantor set. Amer. Math. Monthly 47 (1940), 549-551. · JFM 66.0203.02 · doi:10.2307/2303836 |

[6] | H. Steinhaus: Nowa vlasnośće mnogości G. Cantora. Wektor (1917), 105-107. |

[7] | R.N. Sen: A Course of Geometry. Calcutta University, pp. 19. · Zbl 0052.16401 |

[8] | W. Utz: The distance set of the Cantor dicontinuum. Amer. Math. Monthly 58 (1951), 407-408. · Zbl 0043.05402 · doi:10.2307/2306554 |

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