On cardinal characteristics associated with the strong measure zero ideal

Abstract

Let $\mathcal{S}\mathcal{N}$ be the strong measure zero $\sigma$-ideal. We prove a result providing bounds for $\mathrm{cof}(\mathcal{S}\mathcal{N})$ which implies Yorioka's characterization of the cofinality of the strong measure zero. In addition, we use forcing matrix iterations to construct a model of ZFC that satisfies $\mathrm{add}(\mathcal{S}\mathcal{N})=\mathrm{cov}(\mathcal{S}\mathcal{N})<\mathrm{non}(\mathcal{S}\mathcal{N})<\mathrm{cof}(\mathcal{S}\mathcal{N})$.