  We answer a question of Darji and Keleti by proving in $ZFC$ that there
exists a compact nullset $C_0\subset\RR$ such that for every perfect set
$P\subset\RR$ there exists $x\in\RR$ such that $(C_0+x)\cap P$ is uncountable.
Using this $C_0$ we answer a question of Gruenhage by showing that it is
consistent with $ZFC$ that less than $2^\omega$ many translates of a compact
nullset cover $\RR$.
