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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.
Márton Elekes, and Juris Steprāns. "Less than $2^{ω}$ many translates of a compact nullset may cover the real line." Fundamenta Mathematicae 181.1 (2004): 89-96. <http://eudml.org/doc/286453>.
@article{MártonElekes2004, abstract = {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 $cof() < 2^\{ω\}$) that less than $2^\{ω\}$ many translates of a compact set of measure zero can cover ℝ.}, author = {Márton Elekes, Juris Steprāns}, journal = {Fundamenta Mathematicae}, keywords = {compact subset; measure zero; translate; cover; continuum; perfect set; consistence}, language = {eng}, number = {1}, pages = {89-96}, title = {Less than $2^\{ω\}$ many translates of a compact nullset may cover the real line}, url = {http://eudml.org/doc/286453}, volume = {181}, year = {2004}, }
TY - JOUR AU - Márton Elekes AU - Juris Steprāns TI - Less than $2^{ω}$ many translates of a compact nullset may cover the real line JO - Fundamenta Mathematicae PY - 2004 VL - 181 IS - 1 SP - 89 EP - 96 AB - 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 $cof() < 2^{ω}$) that less than $2^{ω}$ many translates of a compact set of measure zero can cover ℝ. LA - eng KW - compact subset; measure zero; translate; cover; continuum; perfect set; consistence UR - http://eudml.org/doc/286453 ER -