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In the paper we would like to pay attention to some analogies between Haarmeager 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 Polishgroup. 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, andapply our generalization of Piccard's theorem to prove that each $D$-measurablehomomorphism is continuous. Our results refer to the papers \cite{Ch},\cite{Darji} and \cite{FS}.

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups