Cardinal invariants of Haar null and Haar meager sets

Abstract

A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $\mu$ and a Borel set $B$ containing $X$ such that $\mu (gBh)=0$ for every $g,h\in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f:K\to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h\in G$. We calculate (in $ZFC$) the four cardinal invariants ($\mathrm{add}$, $\mathrm{cov}$, $\mathrm{non}$, $\mathrm{cof}$) of these two $\sigma$-ideals for the simplest non-locally compact Polish group, namely in the case $G={\mathbb{Z}}^{\omega }$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Z. Vidnyánszky.