Properties of simple density ideals

Abstract

Let G consist of all functions g:ω→[0,∞) with g(n)→∞ and ng(n)↛0. Then for each g∈G the family Zg={A⊆ω:limn→∞⁡card(A∩n)g(n)=0} is an ideal associated to the notion of so-called upper density of weight g. Although those ideals have recently been extensively studied, they do not have their own name. In this paper, for Reader's convenience, we propose to call them simple density ideals. We partially answer [16, Problem 5.8] by showing that every simple density ideal satisfies the property from [16, Problem 5.8] (earlier the only known example was the ideal Z of sets of asymptotic density zero). We show that there are c many non-isomorphic (in fact even incomparable with respect to Katětov order) simple density ideals. Moreover, we prove that for a given A⊆G with card(A)<b one can construct a family of cardinality c of pairwise incomparable (with respect to inclusion) simple density ideals which additionally are incomparable with all Zg for g∈A. We show that this cannot be generalized to Katětov order as Z is maximal in the sense of Katětov order among all simple density ideals. We examine how many substantially different functions g can generate the same ideal Zg – it turns out that the answer is either 1 or c (depending on g).