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In a previous paper by the first two authors, a tube formula for fractalsprays was obtained which also applies to a certain class of self-similarfractals. The proof of this formula uses distributional techniques and requiresfairly strong conditions on the geometry of the tiling (specifically, the innertube formula for each generator of the fractal spray is required to bepolynomial). Now we extend and strengthen the tube formula by removing theconditions on the geometry of the generators, and also by giving a proof whichholds pointwise, rather than distributionally.  Hence, our results for fractal sprays extend to higher dimensions thepointwise tube formula for (1-dimensional) fractal strings obtained earlier byLapidus 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 rangesover the complex dimensions of the spray, that is, over the poles of thegeometric zeta function of the underlying fractal string and the integers$0,1,...,d$. The resulting "fractal tube formulas" are applied to the importantspecial case of self-similar tilings, but are also illustrated in othergeometrically natural situations. Our tube formulas may also be seen as fractalanalogues of the classical Steiner formula.

Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators