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.
"Geometric cardinal invariants, maximal functions and a measure theoretic pigeonhole principle." Bull. Symbolic Logic 11 (4) 517 - 525, December 2005. https://doi.org/10.2178/bsl/1130335207