A big symmetric planar set with small category projections
Fundamenta Mathematicae, Tome 178 (2003) no. 3, p. 237.

Voir la notice de l'article dans European Digital Mathematics Library

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.
Classification : 03E35, 03E50
Mots-clés : -category projections, transfinite induction, nowhere meager sets, Covering Property Axiom CPA, -oracle 
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Krzysztof Ciesielski; Tomasz Natkaniec. A big symmetric planar set with small category projections. Fundamenta Mathematicae, Tome 178 (2003) no. 3, p. 237. https://geodesic-test.mathdoc.fr/item/FUNDAM_2003__178_3_283346/