Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.


Can $H$ always be covered by a countable family of nowhere dense sets $H\_n$ such that $H\_n^2$ is still nowhere dense, for each $n$?


Clearly, we can assume that $H$ is dense in $G$. I believe it holds when $H$ is (cointained in) a meagre $F\_\sigma$ subgroup, because then we can just take for $H\_n$ the closed nowhere dense sets which add up to $H$, and then use the fact that a meager closed set is nowhere dense.


On the other hand, just taking for $H\_n$ a family of closed nowhere dense sets covering $H$ is not good enough -- for instance if any $H\_n=H\_n^{-1}$ is not null, its square will have nonempty interior by Steinhaus theorem.


I don't have much experience with abstract compact groups, so I have hard time even imaging $H,G$ which do not satisfy the assumptions of the previous paragraph...



Meager subgroups of compact groups
This problem was been answered in negative by M.Laczkovich (<http://www.ams.org/journals/proc/1998-126-06/S0002-9939-98-04241-5/S0002-9939-98-04241-5.pdf>). He constructed a proper Borel subgroup $H$ of the real line which cannot be covered by countably many sets $H\_i$ with nowhere dense sums $H\_i+H\_i$. 


On the other hand, Laczkovich proved that each non-open analytic subgroup $H$ of a Polish locally compact group $G$ can be covered by countably many closed sets of Haar measure zero. 


Trying to generalize this result to non-locally compact groups, Laczkovich proved that any non-open analytic subgroup of a Polish group $G$ belongs to the sigma-ideal generated by the family $\mathcal F$ consisting of closed sets $A$ such that any non-empty open subspace of $A$ contains two relatively open non-empty sets $U,V$ with nowhere dense sum $U+V$ and difference $U-V$. Truly speaking, this result of Laczkovich is not quite satisfactory as each closed subset of $G$ containing a dense set of isolated points belongs to the family $\mathcal F$. Consequently, the $\sigma$-ideal generated by the family $\mathcal F$ coincides with the ideal of meager subsets of $G$.


But we can ask another problem: can each non-open analytic subgroup of a Polish Abelian group be covered by countably many closed Haar null subsets?


