On Some Families of Thin Sets

Lev Bukovský

The work on this research has been supported by the grant 1/7555/20 of Slovenská grantová agentúra VEGA.

Let f, g : ⟨0, 1⟩ ⟶ ⟨, ∞) be continuous, f(0) = g(0) = f(1) = g(1) = 0. Replacing the sinus function by f the families of pseudo-Dirichlet sets pD_{f}, Arbault sets A_{f} and N_{0f}-sets N_{0f} can be defined (see e.g. [BL]) in similar way as the trigonometric ones. For a big family of functions f, g we shall present sufficient and/or necessary conditions for the inclusions N_{0f} ⊆ N_{0g}, N_{0f} ⊆ A_{g}, A_{f} ⊆ A_{g}. Consequently, there exists a large hierarchy of those families.

For a family F of thin sets, the family Prm(F) of F-permitted sets is defined as

A ∈ Prm(F) ≡ (∀B ∈ F)A ∪ B ∈ F.

We show that the estimates for several cardinal characteristics of the trigonometric families of permitted sets obtained in [KK], [RM] holds true for Prm(F), F = pD_{f}, N_{0f}, A_{f}, assuming that f(x − y) ≤ f(x) + f(y), e.g.

non(perfectly measure zero) ≤ non(Prm(F) ≤ min{non(L), non(K)}

and cov(F) ≤ 𝔩 for F = pD_{f}, N_{0f}, A_{f}.

[BL] Bukovský L., Thin sets in a general setting, Tatra Mt. Math. Publ. 14 (1998), 241-260.

[KK] Kada M. and Kamo S., New cardinal invariants related to pseudo–Ditrichlet sets, preprint

[RM] Repický M., A family of permitted trigonometric thin sets, Proc. Amer. Math. Soc. 125 (1997), 137-144.

Address: Prírodovedecká fakulta, Univerzita P. J. Šafárika, Jesenná 5, 041 54 Košice, Slovakia

E-mail: bukovsky@kosice.upjs.sk