We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from ) that less than many translates of a compact set of measure zero can cover ℝ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-1-4,
author = {M\'arton Elekes and Juris Stepr\=ans},
title = {Less than $2^{$\omega$}$ many translates of a compact nullset may cover the real line},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {89-96},
zbl = {1095.28005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-1-4}
}
Márton Elekes; Juris Steprāns. Less than $2^{ω}$ many translates of a compact nullset may cover the real line. Fundamenta Mathematicae, Tome 184 (2004) pp. 89-96. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-1-4/