Generalized kinds of density and the associated ideals

Abstract

We consider modified notions of natural density of subsets of \({\omega:=\{{0,1,\ldots}\}}\). Namely, we introduce the density of the weight \({g:\omega\to [0,\infty)}\) where \({g(n)\to\infty}\) and \({n/g(n)\nrightarrow 0}\). Denote by G the set of all such functions g. We study the associated ideals \({\mathcal{Z}_{g}}\), \({g\in G}\), of sets \({A\subset\omega}\) with g-density zero. We show that \({\mathcal{Z}_{g}}\) is an \({F_{\sigma\delta}}\) P-ideal on \({\omega}\). We give sufficient conditions for proper inclusions of type \({{\mathcal{Z}_{g1}}\varsubsetneq \mathcal{Z}_{g2}}\). Also, we show examples where no inclusion between \({\mathcal{Z}_{g1}}\) and \({\mathcal{Z}_{g2}}\) holds. Finally, we prove that each ideal \({\mathcal{Z}_g}\), \({g\in G}\), is a density ideal in the sense of Farah, and we infer that it cannot be of type \({F_\sigma}\). It turns out that there is no inclusion between \(\{{\mathcal{Z}_g: g\in G}\}\) and the set of all Erdős–Ulam ideals.