Exact Hausdorff measure and intervals of maximum density for Cantor sets

Abstract:

Consider a linear Cantor set $K$, which is the attractor of a linear iterated function system (i.f.s.) $S_{j}x = \rho _{j}x+b_{j}$, $j = 1,\ldots ,m$, on the line satisfying the open set condition (where the open set is an interval). It is known that $K$ has Hausdorff dimension $\alpha$ given by the equation $\sum ^{m}_{j=1} \rho ^{\alpha }_{j} = 1$, and that $\mathcal {H}_{\alpha }(K)$ is finite and positive, where $\mathcal {H}_{\alpha }$ denotes Hausdorff measure of dimension $\alpha$. We give an algorithm for computing $\mathcal {H}_{\alpha }(K)$ exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When $\rho _{1} = \rho _{m}$ (or more generally, if $\log \rho _{1}$ and $\log \rho _{m}$ are commensurable), the algorithm also gives an interval $I$ that maximizes the density $d(I) = \mathcal {H}_{\alpha }(K \cap I)/|I|^{\alpha }$. The Hausdorff measure $\mathcal {H}_{\alpha }(K)$ is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters $\rho _{j}$, it is possible to choose the translation parameters $b_{j}$ in such a way that $\mathcal {H}_{\alpha }(K) = |K|^{\alpha }$, so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.