Full measure universality for Cantor Sets

We investigate variants of the Erdos similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known fact that sets of positive Hausdorff dimension are not measure universal. We prove a weaker result for all Cantor sets A: there is a dense G_ set of full measure Xᵈ, such that for any bi-Lipschitz function f: Rᵈ Rᵈ, the set of translations t such that f (A) +t X is of measure zero. Equivalently, there is a null set Bᵈ such that Rᵈ (f (A) +B) is null for all bi-Lipschitz functions f.