Surface dimension, tiles, and synchronizing automata

We study the surface regularity of compact sets G ⊆ R n which is equal to the supremum of numbers s ≥ 0 such that the measure of the set G∊ G does not exceed C ∊ s , ∊ > 0, where G∊ denotes the ∊-neighborhood of G. The surface dimension is by definition the difference between n and the surface regularity. Those values provide a natural characterization of regularity for sets of positive measure. We show that for self-affine attractors and tiles those characteristics are explicitly computable. We find them for some popular tiles. This, in particular, gives a refined regularity scale for the multivariate Haar wavelets. The classification of attractors of the highest possible regularity is addressed. The relation between the surface regularity and the Holder regularity of multivariate refinable functions and wavelets is found. Finally, the surface regularity is applied to the theory of synchronizing automata, where it corresponds to the concept of parameter of synchronization.