More on the density zero ideal

TL;DR

This paper improves bounds on the cardinal invariants related to the density zero ideal, introducing new variants of the splitting number and exploring their relationships.

Contribution

It provides improved bounds on the cardinal invariants of the density zero ideal and introduces new variants of the splitting number with established relationships.

Findings

Improved upper bound on cov*({Z}_0)

Better lower bound on non*({Z}_0)

New variants of the splitting number and their relationships

Abstract

The main result of this paper is an improvement of the upper bound on the cardinal invariant ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0})$ that was discovered by Raghavan and Shelah in an earlier paper. Here ${\mathcal{Z}}_{0}$ is the ideal of subsets of the set of natural numbers that have asymptotic density zero. This improved upper bound is also dualized to get a better lower bound on the cardinal ${\mathord{\mathrm{non}}}^{\ast}({\mathcal{Z}}_{0})$. En route some variations on the splitting number are introduced and several relationships between these variants are proved.