Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb R$?
This question can be equivalently refolmulated as a question about the ZFC-example of a subgroup $H\subset\mathbb R$ that belongs to the family $(\mathcal M\cap\mathcal N)\setminus\mathcal E$.
Here $\mathcal M,\mathcal N$ and $\mathcal E$ stand for the $\sigma$-ideals of meager sets, sets of measures zero, and sets that can be covered by countably many closed sets of measure zero in $\mathbb R$.
Partial Answers:
Under $cov(\mathcal N)=non(\mathcal M)$ the real line contains a subgroup $H\in\mathcal M\setminus\mathcal E$.
Under $cov(\mathcal N)=cov(\mathcal M)=cof(\mathcal M)$ (in particular, under Martin's Axiom) the real line contains a subgroup $H\in(\mathcal M\cap\mathcal N)\setminus\mathcal E$.
No quasi-analytic subgroup $H\subset\mathbb R$ belongs to the family $\mathcal M\setminus\mathcal E$. A topological space is quasi-analytic if it is a continuous image of a hereditarily Baire metrizable separable space. A space $X$ is hereditarily Baire if each closed subspace of $X$ is Baire. Since each Polish space if hereditarily Baire, each analytic space is quasi-analytic. Because of that (3) implies:
Theorem (Laczkovich, 1998). Each proper analytic subgroup of the real line belongs to the $\sigma$-ideal $\mathcal E$.
Theorem (Talagrand, 1980, Roslanowski & Shelah, 2016). The Cantor cube $\mathbb Z_2^\omega$ contains a subgroup (actually, an ideal on $\omega$) which is not meager but has measure zero.
Theorem (Burke, 1991; Pawlikowski, 1995, Roslanowski & Shelah, 2016). It is consistent that every meager subgroup of $\mathbb R$ has zero measure.
So, we ask about the consistency of a stronger fact: Is it consistent that each meager subgroup of the real line can be covered by countably many closed sets of measure zero?
More information on the items (1)-(3) can be found in this preprint.
In this preprint we construct an example (6.3) of a meager Borel subgroup $H$ in the Polish group $G=\mathbb Z^\omega$ such that $H$ is meager and Haar-null but $H$ cannot be covered by countably many closed Haar-null sets in $G$. This example shows that the 'non-locally compact' version of the above problem has a `Borel' affirmative ZFC-answer. This example also gives a negative answer to the problem (Can each non-open analytic subgroup of a Polish abelian group be covered by countably many closed Haar null subsets?).
