Surface dimension, tiles, and synchronising automata

We study the surface regularity of compact sets $G\subset R^n$ which is equal to the supremum of numbers $s\ge 0$ such that the measure of the set $G_{\epsilon }\setminus G$ does not exceed $C\varepsilon^{s}, \varepsilon > 0$,where$G_{\varepsilon}$denotesthe$\varepsilon$-neighbourhoodof$G$. The surface dimension is by definition the difference between~$n$ and the surface regularity. Those values provide a natural characterisation of regularity for sets of positive measure. We show that for self-affine attractors and tiles those characteristics are explicitly computable and 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 H"older regularity of multivariate refinable functions and wavelets is found. Finally, the surface regularity is applied to the theory of synchronising automata, where it corresponds to the concept of parameter of synchronisation.