Ganguly, D. K.; Majumdar, M. On some properties of the Cantor set. (English) Zbl 0879.26003 Czech. Math. J. 46, No. 3, 553-557 (1996). 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. 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 Keywords:Cantor set; cross ratio × Cite Format Result Cite Review PDF Full Text: DOI EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.