  In a previous paper by the first two authors, a tube formula for fractal
sprays was obtained which also applies to a certain class of self-similar
fractals. The proof of this formula uses distributional techniques and requires
fairly strong conditions on the geometry of the tiling (specifically, the inner
tube formula for each generator of the fractal spray is required to be
polynomial). Now we extend and strengthen the tube formula by removing the
conditions on the geometry of the generators, and also by giving a proof which
holds pointwise, rather than distributionally.
  Hence, our results for fractal sprays extend to higher dimensions the
pointwise tube formula for (1-dimensional) fractal strings obtained earlier by
Lapidus and van Frankenhuijsen.
  Our pointwise tube formulas are expressed as a sum of the residues of the
"tubular zeta function" of the fractal spray in $\mathbb{R}^d$. This sum ranges
over the complex dimensions of the spray, that is, over the poles of the
geometric zeta function of the underlying fractal string and the integers
$0,1,...,d$. The resulting "fractal tube formulas" are applied to the important
special case of self-similar tilings, but are also illustrated in other
geometrically natural situations. Our tube formulas may also be seen as fractal
analogues of the classical Steiner formula.
