Weak Tiling by Unions of Non-overlapping Unit Cubes

Abstract: We study weak tiling by finite unions of pairwise non-overlapping unit cubes in $\mathbb R<sup>d$ that are not necessarily axis-parallel. We obtain a weak-tiling analogue of Keller's classical theorem, which gives a structural restriction on the support of any weak tiling measure associated with the unit cube. This allows us to derive geometric restrictions on configurations of cubes whose union admits a weak tiling. As an application, we prove Fuglede's conjecture for unions of three non-overlapping unit squares in $\mathbb R<sup>2$ as well as unions of two non-overlapping axis-parallel unit cubes in any dimension.