Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

Abstract: Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$\mathcal I$ set in a Polish group. Here $\mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $B\subset X$ of a Polish group $X$ is called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that $f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically Haar-$\mathcal I$ if the set of witness functions ${f\in C(K,X):\forall x\in X\;\;f^{-1}(B+x)\in\mathcal I}$ is comeager in the function space $C(K,X)$. We study (generically) Haar-$\mathcal I$ sets in Polish groups for many concrete and abstract ideals $\mathcal I$, and construct the corresponding distinguishing examples. We prove some results on Borel hull of Haar-$\mathcal I$ sets, generalizing results of Solecki, Elekes, Vidny\'anszky, Dole\v{z}al, Vlas\v{a}k on Borel hulls of Haar-null and Haar-meager sets. Also we establish various Steinhaus properties of the families of (generically) Haar-$\mathcal I$ sets in Polish groups for various ideals $\mathcal I$.