How many closed measure zero sets are needed to cover the real line, really? This is a refinement of an earlier question.
This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
For the reader's convenience, I reproduce below the relevant parts.
Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It is known that
$$\operatorname{cov}(\mathcal{N})\cdot\operatorname{cov}(\mathcal{M})\le 
\operatorname{cov}(\mathcal{E})\le 
\min\{\operatorname{cov}(\mathcal{N})\cdot \mathfrak{d},\mathfrak{r}\}$$
and that the first inequality is consistently strict
(Bartoszynski-Shelah, Ashutosh).
Is the second inequality consistently strict?
 A: Mathias model (i.e., the countable support iteration of length $\omega_2$ of Mathias poset over a model of CH) satisfies $\mathrm{cov}(\mathcal{E})<\mathfrak{b}$ (recall that both $\mathfrak{d}$ and $\mathfrak{r}$ are above $\mathfrak{b}$). Refeer to Bartoszynski-Judah book Set Theory: On the structure of the real line for the citations below.
Mathias forcing adds a dominating real (Lemma 7.4.4). On the other hand, Mathias forcing satisfies the Laver property (Section 7.4A). It is enough to show that any poset $\mathbb{P}$ with the Laver property forces that the closed measure zero sets from the ground model covers the reals in the extension.
Let $\dot{x}$ be a name for a real in $\mathbb{P}$ and $p\in\mathbb{P}$. Let $\dot{h}$ be a name for a member of $\prod_{n<\omega}2^{2n}$ such that $\dot{h}(n)$ codes $\dot{x}\upharpoonright I_n$ where $\langle I_n\rangle_{n<\omega}$ is the interval partition with $|I_n|=2n$. By the Laver property, there are $q\leq p$ in $\mathbb{P}$ and $S\in\prod_{n<\omega}\mathcal{P}(2^{2n})$ such that $|S(n)|\leq 2^n$ and $q\Vdash\dot{h}(n)\in S(n)$ for all $n<\omega$. Now, $C_S:=\{z\in 2^\omega:\forall_{n<\omega}(z\upharpoonright I_n\in S(n))\}$ is a closed measure zero set (coded in the ground model) and $q\Vdash\dot{x}\in C_S$ (elements for this argument are taken from Section 2.6A).
Other example is a countable support iteration of length $\omega_2$, over a model of CH, alternating between Miller forcing and a proper poset with the Laver property that increases $\mathfrak{r}$ (e.g. Silver forcing, which has the Sacks property).
I also have some examples with finite support iterations of ccc posets (so large continuum is possible, plus having many invariants with different values), though they involve many technicalities (like preservation theorems) not published so far.
