Less than 2 / omega many translates of a compact nullset may cover the real line
Abstract
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)∩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
ω
many translates of a compact nullset cover $\RR$.