Decomposition of Integral Self-Affine Multi-Tiles
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abstract

In this paper, we propose a method to decompose an integral self-affine
${\mathbb Z}^n$-tiling set $K$ into measure disjoint pieces $K_j$ satisfying
$K=\displaystyle\bigcup K_j$ in such a way that the collection of sets $K_j$
forms an integral self-affine collection associated with the matrix $B$ and
this with a minimum number of pieces $K_j$. When used on a given measurable
$\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$, this decomposition terminates
after finitely many steps if and only if the set $K$ is an integral self-affine
multi-tile. Furthermore, we show that the minimal decomposition we provide is
unique.