We show that under appropriate set-theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager set A ⊂ ℝ such that (i) the set {c ∈ ℝ: π[(f+c) ∩ (A×A)] is not meager} is meager for each continuous nowhere constant function f: ℝ → ℝ, (ii) the set {c ∈ ℝ: (f+c) ∩ (A×A) = ∅} is nowhere meager for each continuous function f: ℝ → ℝ. The existence of such a set also follows from the principle CPA, which holds in the iterated perfect set model. We also prove that the existence of a set A as in (i) cannot be proved in ZFC alone even when we restrict our attention to homeomorphisms of ℝ. On the other hand, for the class of real-analytic functions a Bernstein set A satisfying (ii) exists in ZFC.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-4,
author = {Krzysztof Ciesielski and Tomasz Natkaniec},
title = {A big symmetric planar set with small category projections},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {237-253},
zbl = {1059.03050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-4}
}
Krzysztof Ciesielski; Tomasz Natkaniec. A big symmetric planar set with small category projections. Fundamenta Mathematicae, Tome 177 (2003) pp. 237-253. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-4/