Haar null sets and the consistent reflection of non-meagreness

A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B\supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu (gBh)=0$ for every $g,h\in G$. We prove that there exists a set $X \subset $\mathbb{R}$$thatisnotLebesguenullandaBorelprobabilitymeasure$\mu$suchthat$\mu(X + t) = 0$forevery$t \in $\mathbb{R}$$. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$suchthatforeverynon-meagreset$X \subset G$thereexistsa$t \in G$suchthat$C \cap (X + t)$isrelativelynon-meagrein$C$. This essentially generalises results of Bartoszy'nski and Burke-Miller.