Sets with Large Intersection Properties

For 0 < s ⩽ n let s be the class of G δ ‐subsets of R n such that F ∈ s if∩ i − 1 ∞f i ( F ) has Hausdorff dimension at least s for all sequences of similarity transformations{ f 1 } i − 1 ∞ . We show that s is closed under countable intersections and under bi‐Lipschitz functions, and thus is the maximal class of G δ ‐sets of Hausdorff dimension at least s that is closed under countable intersection and similarities. We also show that sets in s must have packing dimension n . Many examples of s ‐sets occur in Diophantine approximation.