Classification of tile digit sets as product-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 \(pq\).