The closed Steinhaus properties of $σ$-ideals on topological groups

Abstract: We prove that any meager quasi-analytic subgroup of a
topological group $G$ belongs to every $\sigma$-ideal $\mathcal I$ on
$G$ possessing the closed ±n-Steinhaus property for some $n∈\mathbb
N$. An ideal I on a topological group G is defined to have the closed
±n-Steinhaus property if for any closed subsets $A_1,…,A_n∉I$ of G the
product $(A_1∪A^{−1}_1)⋯(A_n∪A^{−1}_n)$ is not nowhere dense in G.
Since the σ-ideal E generated by closed Haar null sets in a locally
compact group G has the closed ±2-Steinhaus property, we conclude that
each meager quasi-analytic subgroup H⊂G belongs to the ideal E. For
analytic subgroups of the real line this result was proved by
Laczkovich in 1998. We shall discuss possible generalizations of the
Laczkovich Theorem to non-locally compact groups and construct an
example of a meager Borel subgroup in $Z^ω$ which cannot be covered by countably many closed Haar-null (or even closed Haar-meager) sets. On the other hand, assuming that cof(M)=cov(M)=cov(N) we construct a
subgroup $H⊂2^ω$ which is meager and Haar null but does not belong to
the σ-ideal E. The construction uses a new cardinal characteristic
$voc^∗(I,J)$ which seems to be interesting by its own.

More details can be found in the preprint http://arxiv.org/pdf/1509.09073v1