Exact Hausdorff measure and intervals of maximum density for Cantor sets

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,..., 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 Sigma(j=1)(m) rho(j)(alpha) = 1, and that H-alpha(K) is finite and positive, where H-alpha denotes Hausdorff measure of dimension alpha. We give an algorithm for computing 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) = H-alpha(K boolean AND I)/\I\(alpha). The Hausdorff measure 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 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.