More about the cofinality and the covering of the ideal of strong measure zero sets

Abstract

We improve the previous work of Yorioka and the first author about the combinatorics of the ideal $\mathrm{SN}$ of strong measure zero sets of reals. We refine the notions of dominating systems of the first author and introduce the new combinatorial principle $\mathrm{DS}\left(\delta \right)$ that helps to find simple conditions to deduce $d_{\kappa }\le \mathrm{cof}\left(\mathrm{SN}\right)$ (where $d_{\kappa }$ is the dominating number on ${\kappa }^{\kappa }$). In addition, we find a new upper bound of $\mathrm{cof}\left(\mathrm{SN}\right)$ by using products of relational systems and cardinal characteristics associated with Yorioka ideals.

In addition, we dissect and generalize results from Pawlikowski to force upper bounds of the covering of $\mathrm{SN}$, particularly for finite support iterations of precaliber posets.

Finally, as applications of our main theorems, we prove consistency results about the cardinal characteristics associated with $\mathrm{SN}$ and the principle $\mathrm{DS}\left(\delta \right)$. For example, we show that $\mathrm{cov}\left(\mathrm{SN}\right)<\mathrm{non}\left(\mathrm{SN}\right)=c<\mathrm{cof}\left(\mathrm{SN}\right)$ holds in Cohen model, and we refine a result (and the proof) of the first author about the consistency of $\mathrm{cov}\left(\mathrm{SN}\right)<\mathrm{non}\left(\mathrm{SN}\right)<\mathrm{cof}\left(\mathrm{SN}\right)$, with $c$ in any desired position with respect to $\mathrm{cof}\left(\mathrm{SN}\right)$, and the improvement that $\mathrm{non}\left(\mathrm{SN}\right)$ can be singular here.