Erdős–Ulam ideals vs. simple density ideals

Abstract

The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function g:ω→[0,∞) such that limn→∞⁡g(n)=∞ and ng(n) does not converge to 0. Then the family Zg={A⊆ω:limn→∞⁡card(A∩n)g(n)=0} is an ideal called simple density ideal (or ideal associated to upper density of weight g). We compare this class of ideals with Erdős–Ulam ideals. In particular, we show that there are ⊑-antichains of size c among Erdős–Ulam ideals which are and are not simple density ideals (in [12] it is shown that there is also such an antichain among simple density ideals which are not Erdős–Ulam ideals). We characterize simple density ideals which are Erdős–Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erdős–Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal I, asserts that given B∈I and C⊆ω with card(C∩n)≤card(B∩n) for all n, we have C∈I. This notion is inspired by [3] and is later applied in [12] for a partial solution of [15, Problem 5.8]. Finally, we pose some open problems.