Billingsley dimension on shift spaces

We consider a class of subshifts Σ over a finite alphabet, including sofic shifts. For a large class of metrics we determine the Hausdorff dimension of sets of points of Σ defined by their limit-point set under the empirical measure. Our approach to computing the Billingsley dimension of saturated sets is fundamentally different and applies to more general shift spaces and measures than the technique of Billingsley, which was significantly developed by Cajar and recently extended by others. One main feature of our approach is an algorithmic construction of a large (in the sense of dimension theory) subset of a saturated set. This generalizes similar constructions of subsets of normal or generic points.