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

TL;DR

This paper extends pointwise tube formulas for fractal sprays and self-similar tilings to arbitrary generators, removing previous geometric restrictions and generalizing earlier results to higher dimensions with applications to fractal geometry.

Contribution

It provides a generalized, pointwise tube formula for fractal sprays and tilings, removing geometric constraints on generators and extending prior 1D results to higher dimensions.

Findings

Tube formulas expressed as sums of residues of tubular zeta functions.

Formulas apply to self-similar tilings and other natural geometries.

Extension of fractal string results to higher dimensions.

Abstract

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.