By a result of Laczkovich ('Analytic subgroups of the reals' Proc AMS Vol 126 (1998)), any non-open analytic subgroup of a Polish locally compact group can be covered by countably many closed Haar null sets.

Is this result of Laczkovich true for any Polish (not necessarily locally compact) groups? More precisely:

**Problem 1:** Does each non-open analytic subgroup $H$ of a Polish abelian group $G$ belong to the $\sigma$-ideal $\mathcal E$ generated by closed Haar null subsets of $G$?

A Borel subset $A$ of a Polish group $G$ is Haar null of there exists a Borel probability measure $\mu$ on $G$ such that $\mu(xAy)=0$ for all $x,y\in G$.

A related weaker question also seems to be open.

**Problem 2:** Does each non-open analytic subgroup $H$ of a Polish abelian group $G$ belong to the $\sigma$-ideal $\mathcal M_h$ generated by closed Haar-meager subsets of $G$?

A Borel subset $A$ of a topological group $G$ is called *Haar-meager* if there exists a continuous map $f:K\to G$ defined on a compact metrizable space $K$ such that $f^{-1}(xAy)$ is meager in $G$ for all $x,y\in G$. It can be shown that a closed subset $A$ of a topological group $G$ is Haar-meager if and only if there exists a compact subset $K\subset G$ such that for every $x,y\in G$ the intersection $K\cap xAy$ is nowhere dense in $K$. It is easy to see that each closed Haar-null set is Haar-meager. More information on Haar meager sets can be found in the paper "On Haar meager sets" by Darji.