Classification of tile digit sets as product-forms

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