Real Analysis Exchange

Cardinality of bases of families of thin sets.

Lev Bukovský

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 9 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149860206

Mathematical Reviews number (MathSciNet)
MR2061300

Zentralblatt MATH identifier
1060.03069

Subjects
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

Keywords
trigonometric thin sets family of thin sets basis tower

Citation

Bukovský, Lev. Cardinality of bases of families of thin sets. Real Anal. Exchange 29 (2003), no. 1, 147--153. https://projecteuclid.org/euclid.rae/1149860206


Export citation