Decomposition of integral self‐affine multi‐tiles

In contrast to the situation with self‐affine tiles, the representation of self‐affine multi‐tiles may not be unique (for a fixed dilation matrix). Let K⊂Rn be an integral self‐affine multi‐tile associated with an n×n integral, expansive matrix B and let K tile Rn by translates of Zn. In this work, we propose a stepwise method to decompose K into measure disjoint pieces Kj satisfying K=⋃Kj in such a way that the collection of sets Kj forms an integral self‐affine collection associated with the matrix B and this with a minimum number of pieces Kj. When used on a given measurable subset K which tiles Rn by translates of Zn, 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.