Splitting for Integer Tilings

Abstract We consider translational integer tilings by finite sets $A\subset{\mathbb{Z}}$. We introduce a new method based on splitting, together with a new combinatorial interpretation of some of the main tools of [23] and [24]. We also use splitting to prove the Coven–Meyerowitz conjecture for a new class of tilings $A\oplus B={\mathbb{Z}}_{M}$. This includes tilings of period $M=p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$ with $p_{1}>p_{2}^{n_{2}-1}p_{3}^{n_{3}-1}$, and tilings of period $M=p_{1}^{n_{1}}p_{2}^{2}p_{3}^{2}p_{4}^{2}$ with $p_{1}>p_{2}p_{3}p_{4}$, where $p_{1},p_{2},p_{3},p_{4}$ are distinct primes and $n_{1},n_{2},n_{3}\in{\mathbb{N}}$.