Small sets of reals through the prism of fractal dimensions

TL;DR

This paper explores special classes of sets in metric spaces characterized by their fractal dimensions, establishing equivalences among various null sets in the Cantor space and their properties under continuous functions.

Contribution

It introduces and analyzes new classes of null sets based on fractal dimensions, showing their equivalences and invariance properties in the Cantor space.

Findings

In 2^ω, H-null, upper H-null, directed P-null, and P-null sets coincide with strongly null, meager-additive, T' and null-additive sets.

A subset of 2^ω is meager-additive if and only if it is E-additive.

Continuous images of meager-additive and null-additive sets preserve these properties.

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.