Tube Formulas and Complex Dimensions of Self-Similar Tilings

We extend some aspects of the theory of fractal strings and their complex dimensions from the real line to general Euclidean spaces. This is accomplished by using the explicit formulas and techniques of [La-vF4]. We use the self-similar tilings constructed in [Pe1] to define a zeta function which encodes the scaling properties of the tiling. This allows us to define a generating function for the geometry of the tiling as a more complicated geometric zeta function. The complex dimensions of the tiling are defined to be the poles of the geometric zeta function. We obtain a tube formula for a self-similar tiling in R as a power series in ε. The Steiner formula gives the volume of the ε-neighbourhood of a compact convex subset A ⊆ R, as a polynomial in ε with coefficients given by the curvature measures of A, summed over i = 0, 1, . . . , d − 1. We obtain a fractal extension of this, in which the sum is not only taken over the integers 0, 1, . . . , d − 1, but also has a term for each complex dimension and thus has infinitely many terms in general. This provides further justification for the term “complex dimension”. It also extends to higher dimensions and sheds new light on the tube formula for fractals strings obtained in [La-vF4]. Received by the editor May 25, 2006. 2000 Mathematics Subject Classification. Primary: 11M41, 28A12, 28A75, 28A80, 52A39, 52C07. Secondary: 11M36, 28A78, 28D20, 42A16, 42A75, 52A20, 52A38.