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 subset of R-n be an integral self-affine multi-tile associated with an n x n integral, expansive matrix B and let K tile R-n by translates of Z(n). In this work, we propose a stepwise method to decompose K intomeasure disjoint pieces K-j satisfying K = boolean OR K-j in such a way that the collection ofsets K-j forms an integral self-affine collection associated with the matrix B and thiswith a minimum number of pieces K-j. When used on a given measurable subset K which tiles R-n by translates of Z(n), this decomposition terminates after finitely manysteps if and only if the set K is an integral self-affine multi-tile. Furthermore, we showthat the minimal decomposition we provide is unique.