The main result of this paper is an improvement of the upper bound on the cardinal invariant ${\mathrm{cov}}^{⁎}({\mathcal{Z}}_0)$ that was discovered in [11]. 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 ${\mathrm{non}}^{⁎}({\mathcal{Z}}_0)$. En route some variations on the splitting number are introduced and several relationships between these variants are proved.

中文翻译：

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密度零理想和分裂数

本文的主要结果是改进了基不变量的上限 ${\mathrm{冠状病毒}}^{⁎}（{\mathcal{ž}}_0）$在[11]中发现的。这里${\mathcal{ž}}_0$是具有零渐近密度的自然数集的理想子集。改进的上限也被双重化，从而在基数上获得更好的下限${\mathrm{非}}^{⁎}（{\mathcal{ž}}_0）$。在途中介绍了分割数的一些变体，并证明了这些变体之间的几种关系。