  We use the self-similar tilings constructed by the second author in
"Canonical self-affine tilings by iterated function systems" to define a
generating function for the geometry of a self-similar set in Euclidean space.
This tubular zeta function encodes scaling and curvature properties related to
the complement of the fractal set, and the associated system of mappings. This
allows one to obtain the complex dimensions of the self-similar tiling as the
poles of the tubular zeta function and hence develop a tube formula for
self-similar tilings in \$\mathbb{R}^d$. The resulting power series in
$\epsilon$ is a fractal extension of Steiner's classical tube formula for
convex bodies $K \ci \bRd$. Our sum has coefficients related to the curvatures
of the tiling, and contains terms for each integer $i=0,1,...,d-1$, just as
Steiner's does. However, our formula also contains terms for each complex
dimension. This provides further justification for the term "complex
dimension". It also extends several aspects of the theory of fractal strings to
higher dimensions and sheds new light on the tube formula for fractals strings
obtained in "Fractal Geometry and Complex Dimensions" by the first author and
Machiel van Frankenhuijsen.
