On Cantor sets and doubling measures

TL;DR

This paper investigates conditions under which Cantor sets on the real line have positive or null measure for all doubling measures, including atomic measures on specific midpoint Cantor sets.

Contribution

It provides necessary and sufficient conditions for Cantor sets to have positive or null measure under all doubling measures, extending to atomic measures on midpoint Cantor sets.

Findings

Characterizes when Cantor sets have positive measure for all doubling measures.

Identifies conditions for null measure for all doubling measures.

Analyzes atomic doubling measures on midpoint Cantor sets.

Abstract

For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for atomic doubling measures defined on certain midpoint Cantor sets.