A big symmetric planar set with small category projections
Volume 178 / 2003
Fundamenta Mathematicae 178 (2003), 237-253
MSC: Primary 03E35; Secondary 26A99, 03E50.
DOI: 10.4064/fm178-3-4
Abstract
We show that under appropriate set-theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager set $A\subset{\mathbb R}$ such that
(i) the set $\{c\in{\mathbb R}\colon\, \pi[({f+c}) \cap (A\times A)]\hbox{ is not meager}\}$ is meager for each continuous nowhere constant function $f\colon\,{\mathbb R}\to{\mathbb R}$,
(ii) the set $\{c\in{\mathbb R}\colon\, (f+c)\cap (A\times A)=\emptyset\}$ is nowhere meager for each continuous function $f\colon\,{\mathbb R}\to{\mathbb R}$.
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 $\mathbb R$. On the other hand, for the class of real-analytic functions a Bernstein set $A$ satisfying (ii) exists in ZFC.
