Real Analysis Exchange

Cardinality of bases of families of thin sets.

Lev Bukovský

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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 trigonometric families $\mathcal{D}$, $\mathcal{pD}$, $\mathcal{B}_0$, $\mathcal{N}_0$, $\mathcal{B}$, $\mathcal{N}$, $\mathcal{wD}$ and $\mathcal{A}$ has cardinality at least $\mathfrak c$. Moreover, we construct an increasing tower of pseudo Dirichlet sets of cardinality $\mathfrak t$.

Article information

Real Anal. Exchange, Volume 29, Number 1 (2003), 147-153.

First available in Project Euclid: 9 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E05: Other combinatorial set theory 42A20: Convergence and absolute convergence of Fourier and trigonometric series 03E75: Applications of set theory 42A28 26A99: None of the above, but in this section

trigonometric thin sets family of thin sets basis tower


Bukovský, Lev. Cardinality of bases of families of thin sets. Real Anal. Exchange 29 (2003), no. 1, 147--153.

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