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Cardinality of bases of families of thin sets. (English) Zbl 1060.03069

Summary: We construct a family of Dirichlet sets of cardinality \({\mathfrak c}\) such that the arithmetic sum of any two members of the family contains an open interval. As a corollary we obtain that every basis of many families of thin sets has cardinality at least \({\mathfrak c}\). Especially, every basis of any of the trigonometric families \({\mathcal D}\), \(p {\mathcal D}\), \({\mathcal B}_0\), \({\mathcal N}_0\), \({\mathcal B}\), \({\mathcal N}\), \(w{\mathcal D}\) and \({\mathcal A}\) has cardinality at least \({\mathfrak c}\). Moreover, we construct an increasing tower of pseudo-Dirichlet sets of cardinality \({\mathfrak t}\).

MSC:

03E05 Other combinatorial set theory
42A20 Convergence and absolute convergence of Fourier and trigonometric series
03E75 Applications of set theory
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
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