Small sets of reals through the prism of fractal dimensions

Abstract

A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff dimension. We investigate these sets and show that in 2^\omega{} they coincide, respectively, with strongly null, meager-additive, T' and null-additive sets. Some consequences: A subset of 2^\omega{} is meager-additive if and only if it is E-additive; if f:2^\omega->2^\omega{} is continuous and X is meager-additive, then so is f(X), and likewise for null-additive and T'-sets.