On Cantor sets and doubling measures (Q432427)

A doubling measure \(\mu\) on a metric space is a measure such that there is a constant \(c<\infty\) such that \(0<\mu(B(x,2r))\leq c\cdot\mu(B(x,r))<\infty\) for all~\(x\) and~\(r\). The authors analyze a general construction of Cantor sets and obtain conditions under which these Cantor sets satisfy one of the following four combinations of conditions: measure zero/non-zero for some/all doubling measures on~\(\mathbb{R}\).