A generalization of Chaitin's halting probability $\Omega$ and halting self-similar sets

Abstract

We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D>0. Chaitin's halting probability \Omega is generalized to \Omega^D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree of randomness of each point in Euclidean space through its base-two expansion. It is then shown that the maximum value of such a degree of randomness provides the Hausdorff dimension of a self-similar set that is computable in a certain sense. The class of such self-similar sets includes familiar fractal sets such as the Cantor set, von Koch curve, and Sierpinski gasket. Knowledge of the property of \Omega^D allows us to show that the self-similar subset of [0,1] defined by the halting set of a universal algorithm has a Hausdorff dimension of one.