Decomposition of Integral Self-Affine Multi-Tiles

Suppose a measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with an $n\times n$ integral expansive matrix $B$. We provide an algorithm to decompose $K$ into disjoint pieces $K_j$ which satisfy $K=\displaystyle\bigcup K_j$ in such a way that the collection of sets $K_j$ is an integral self-affine collection associated with the matrix $B$ and the number of pieces $K_j$ is minimal. Using this algorithm, we can determine whether a given measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with any given $n\times n$ integral expansive matrix $B$. Furthermore, we show that the minimal decomposition we provide is unique.