A big symmetric planar set with small category projections

by

Krzysztof Ciesielski and Tomasz Natkaniec

Fund. Math. 178(3) (2003), 237-253.

We show that under appropriate set theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager subset A of R such that

1. for each continuous nowhere constant function f: R-->R the set
{c in R: proj[(f+c)\cap (AxA)] is not meager}
is meager, and
2. for each continuous f: R-->R the set
{c in R: (f+c)\cap (AxA) is empty}
is nowhere meager.
The existence of such a set follows also from the principle CPA, which holds in the iterated perfect set model. We also prove that the existence of a set A as in (1) 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 (2) exists in ZFC.