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

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