# Decomposition of Integral Self-Affine Multi-Tiles Academic Article

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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.