Cardinality of bases of families of thin sets.

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 $\mathfrak t$.