Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  433.04002
Autor:  Erdös, Paul
Title:  Some remarks on subgroups of real numbers. (In English)
Source:  Colloq. Math. 42, 119-120 (1979).
Review:  Stanislaw Hartman asked: Is there a group G of real numbers which is of measure 0and of second category? (Problem of Stanislaw Hartman). In the present note the author gives an affirmative answer to this problem assuming the continuum hypothesis: it is sufficient to consider as an \omega1-sequence An (n < \omega1) the system of all subsets of R which are Fsigma and of first category and then to construct a strictly increasing \omega1-sequence Gn of countable subgroups of (R,+) such that Gn\cap\cupi < nGi = Gr\cap\cupj < nAj for n < r < \omega1; then G: = \cup Gn(n < \omega1) is a requested group. Dually, the author establishes (under CH) the existence of a subgroup of R that is of the first category but not of measure 0. In this connexion let us recall a remarkable result of W.Sierpinski [Fundam. Math. 22, 276-280 (1934; Zbl 009.20405), and also pp. 207-210 in his Oeuvres choisies. Tome III (1976; Zbl 316.01012)]. There is a permutation p in R! which is a 1 -1 mapping between the system of all sets \subset R of measure zero and the system of all subsets of R which are of the first category. P.Erdös [Ann. Math. II. Ser. 44, 643-646 (1943; Zbl 060.13112)] improved this result of Sierpinski answering in affirmative a question of Sierpinski whether moreover p could satisfy p-1 = p. The author states that a similar result holds if one requires that G be a field; in this case the move group ––> ring, field is easy; is such a move possible in the following statement? EV: For every 0 \leq \alpha \leq 1 there is a group of real numbers of Hausdorff dimension \alpha [cf. P.Erdös and B.Volkmann, J. Reine Angew. Math. 221, 203-208 (1966; Zbl 135.10202)].
Reviewer:  \D.Kurepa
Classif.:  * 04A15 Descriptive set theory
                   04A30 Continuum hypothesis and generalizations
                   28A05 Classes of sets
                   54F45 Dimension theory (general topology)
Keywords:  Hausdorff dimension; measure zero; category two; subgroups of real numbers
Citations:  Zbl.009.204; Zbl.316.01012; Zbl.060.131; Zbl.135.102

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