Self‐similarity and zeta functions of multidimensional q ‐automatic sets

Abstract

This article uses methods from discrete fractal zeta functions to describe geometric features of multidimensional ‐automatic sets. Our main result attaches an almost periodic “weighted density” function whose supremum defines a weighted density for any such automatic set. This extends earlier work of Barbé‐von Haeseler. An interesting consequence of our zeta function method is that we can connect geometric properties of the lattice points parameterizing the constituent decimation sequences of a ‐automatic set to the density of such sequences. We illustrate this with three examples. In doing so, we extend the work of Adamscewski–Bell and Allen–Edwards–Harper–Olsen. In addition, we use fractal zeta functions to analyze the weighted density function of two types of “modular digit sets”.