Classification of tile digit sets as product-forms

TL;DR

This paper characterizes digit sets that form self-affine tiles in higher dimensions, extending previous work by linking tile digit sets to integer tiling and product-form structures, especially for matrices with specific prime power factorizations.

Contribution

It proves that tile digit sets in ${f Z}^s$ are integer tiles and explicitly characterizes all such sets for matrices with prime power factorizations as modulo product-forms.

Findings

Tile digit sets in ${f Z}^s$ are integer tiles.

Complete classification of tile digit sets for matrices with $A=p^{eta}q$.

Extension of previous results to more general matrix forms.

Abstract

Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.