Less than $2^{\omega}$ many translates of a compact nullset may cover the real line
Volume 181 / 2004
Fundamenta Mathematicae 181 (2004), 89-96
MSC: Primary 28E15; Secondary 03E17, 03E35.
DOI: 10.4064/fm181-1-4
Abstract
We answer a question of Darji and Keleti by proving that there exists a compact set $C_0\subset\mathbb R$ of measure zero such that for every perfect set $P\subset\mathbb R$ there exists $x\in\mathbb R$ 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 ${\rm cof}({\cal N})<2^\omega$) that less than $2^\omega$ many translates of a compact set of measure zero can cover $\mathbb R$.
