INFINITE PATTERNS THAT CAN BE AVOIDED BY MEASURE

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