Tube formulas and complex dimensions of self-similar tilings

TL;DR

This paper develops a fractal tube formula for self-similar tilings in Euclidean space using a generating function called the tubular zeta function, linking complex dimensions to geometric and curvature properties.

Contribution

It introduces a new tube formula for self-similar tilings that incorporates complex dimensions, extending classical convex geometry results to fractal tilings in higher dimensions.

Findings

Complex dimensions are identified as poles of the tubular zeta function.

The power series in epsilon generalizes Steiner's classical tube formula.

The formula includes terms for each complex dimension, enriching fractal geometry analysis.

Abstract

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.