Exact Hausdorff measure and intervals of maximum density for Cantor sets

Abstract

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