Less than $2^ω$ many translates of a compact nullset may cover the real line

We answer a question of Darji and Keleti by proving that there exists a compact set $C_0\subset\RR$ of measure zero 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 $ZFC$ (as it follows e.g. from $\textrm{cof}(\iN)<2^ω$) that less than $2^ω$ many translates of a compact set of measure zero can cover $\RR$.

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