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Application of covering sets. (English) Zbl 0931.26002

\(A\subset {\mathbb R}\) is called \(\kappa\)-covering set if for each \(X\subset {\mathbb R}\), \(|X|\leq\kappa\), there is a \(t\in{\mathbb R}\) such that \(X+t\subset A\). The author proves several theorems about covering sets which generalize and unify some results about the additive subgroups of the reals and the algebraic difference of sets (see B. King [Real Anal. Exch. 19, No. 2, 478-490 (1994; Zbl 0804.28002)] and K. Muthuvel [Real Anal. Exch. 20, 819-822 (1995; Zbl 0828.26002)]). He shows in particular that the complement of a finite union of proper subgroups of \({\mathbb R}\) is of the size continuum and nowhere meager.

MSC:

26A03 Foundations: limits and generalizations, elementary topology of the line
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E15 Descriptive set theory
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