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**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 \omega_{1}-sequence A_{n} (n < \omega_{1}) the system of all subsets of R which are F_{s}igma and of first category and then to construct a strictly increasing \omega_{1}-sequence G_{n} of countable subgroups of (R,+) such that G_{n}\cap\cup_{i < n}G_{i} = G_{r}\cap\cup_{j < n}A_{j} for n < r < \omega_{1}; then G: = \cup G_{n}(n < \omega_{1}) 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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