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Intervals contained in arithmetic combinations of sets. (English) Zbl 0838.28002

In this paper, the author gets the following result, which is a further extension of a classical one by Steinhaus:
Theorem. Let \(G\) and \(H\) be dense \(G_\delta\) sets in non-empty open intervals \(I\) and \(J\), respectively. If \(\&\) is any one of the four arithmetic operations \(+\), \(-\) \(\cdot\) or \(/\), then \[ G\& H= I\& J, \] except that in the case of multiplication and division 0 might be in \(I\& J\) but not in \(G\& H\).

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54E52 Baire category, Baire spaces
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