Infinite Patterns that can be Avoided by Measure

Abstract

A set A of real numbers is called universal (in measure) if every measurable set of positive measure necessarily contains an affine copy of A . All finite sets are universal, but no infinite universal sets are known. Here we prove some results related to a conjecture of Erdős that there is no infinite universal set. For every infinite set A , there is a set E of positive measure such that ( x + tA )⊆ E fails for almost all (Lebesgue) pairs ( x , t ). Also, the exceptional set of pairs ( x , t ) (for which ( x + tA )⊆ E ) can be taken to project to a null set on the t -axis. Finally, if the set A contains large subsets whose minimum gap is large (in a scale-invariant way), then there is E ⊆ R of positive measure which contains no affine copy of A .