Less than $2^{\omega}$ many translates of a compact nullset may cover the real line

Márton Elekes ¹   ; Juris Steprāns ²

¹ Rényi Alfréd Institute Reáltanoda u. 13-15 Budapest, 1053, Hungary
² Department of Mathematics York University Toronto, Ontario M3J 1P3, Canada

Fundamenta Mathematicae, Tome 181 (2004) no. 1, pp. 89-96

Résumé

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 $\mathbb R$.

Détail

DOI : 10.4064/fm181-1-4

Keywords: answer question darji keleti proving there exists compact set subset mathbb measure zero every perfect set subset mathbb there exists mathbb cap uncountable using answer question gruenhage showing consistent zfc follows cof cal omega omega many translates compact set measure zero cover nbsp mathbb