The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure

Let ${\mathcal{N}}_{\mathrm{\Omega }}^h$ be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of $({\mathcal{N}}_{\mathrm{\Omega }}^h,{\mathcal{N}}_{\mathrm{\Omega }}^h,\subseteq )$ to a localization system with finite coordinate sets. It follows that ${\mathfrak{v}}_{D,g}^{\mathrm{\forall }}\le \mathrm{add}({\mathcal{N}}_{\mathrm{\Omega }}^h)$ and $\mathrm{cof}({\mathcal{N}}_{\mathrm{\Omega }}^h)\le {\mathfrak{c}}_{D,g}^{\mathrm{\forall }}$. Consequently, we prove the consistency of $\mathfrak{d}<\mathrm{add}({\mathcal{N}}_{\mathrm{\Omega }}^h)$ and, separately, $\mathrm{cof}({\mathcal{N}}_{\mathrm{\Omega }}^h)<\mathfrak{b}$. Hence the additivity and cofinality of ${\mathcal{N}}_{\mathrm{\Omega }}^h$ can differ from those of the Lebesgue null ideal.