Geometric cardinal invariants, maximal functions and a measure theoretic pigeonhole principle

Abstract

It is shown to be consistent with set theory that every set of reals of size ℵ₁ is null yet there are ℵ₁ planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.