  In the paper we would like to pay attention to some analogies between Haar
meager sets and Haar null sets. Among others, we will show that $0\in \inn
(A-A)$ for each Borel set $A$, which is not Haar meager in an abelian Polish
group. Moreover, we will give an example of a Borel non-Haar meager set
$A\subset c_0$ such that $\inn (A+A)=\emptyset$. Finally, we will define
$D$-measurability as a topological analog of Christensen measurability, and
apply our generalization of Piccard's theorem to prove that each $D$-measurable
homomorphism is continuous. Our results refer to the papers \cite{Ch},
\cite{Darji} and \cite{FS}.
