$\mathfrak{d}={\omega }_1$ implies $\mathfrak{a}={\omega }_1$

We prove in ZFC that $\mathfrak{d}={\omega }_1$ implies $\mathfrak{a}={\omega }_1$, settling a question of Roitman from the 1970s. Here $\mathfrak{d}$ is the least size of a family that eventually dominates every function in ${\omega }^{\omega }$, and $\mathfrak{a}$ is the least size of an infinite maximal almost disjoint (MAD) family of infinite subsets of $\omega$. Consequently, $\mathfrak{a}\le \mathfrak{d}$ whenever the continuum $\mathfrak{c}=2^{{\mathrm{\aleph }}_0}$ is at most ${\omega }_2$. This answers negatively Shelah’s question whether $\mathfrak{d}<\mathfrak{a}$ is consistent with $\mathfrak{c}={\omega }_2$, and shows that his model of $\mathfrak{d}={\omega }_2<\mathfrak{a}=\mathfrak{c}={\omega }_3$ has the least possible values of $\mathfrak{d}$, $\mathfrak{a}$ and $\mathfrak{c}$ among models of $\mathfrak{d}<\mathfrak{a}$. The proof also gives a preservation theorem: from any dominating family of size ${\omega }_1$, we construct a MAD family of size ${\omega }_1$ that remains maximal in every outer model in which the given family is still dominating. Thus, under $\mathfrak{d}={\omega }_1$, some MAD family is indestructible by every ${\omega }^{\omega }$-bounding forcing; it can moreover be chosen almost strongly separable, hence Cohen-indestructible. These results give negative answers, under $\mathfrak{d}={\omega }_1$, to two questions of Hrušák and to one of Brendle, Guzmán, Hrušák and Raghavan.