Periodicity of joint co-tiles in $\mathbb{Z}^d$

An old theorem of Newman asserts that any tiling of $\mathbb{Z}$ by a finite set is periodic. A few years ago, Bhattacharya proved the periodic tiling conjecture in $\mathbb{Z}^2$. Namely, he proved that for a finite subset $F$ of $\mathbb{Z}^2$, if there exists $A \subseteq \mathbb{Z}^2$ such that $F \oplus A = \mathbb{Z}^2$ then there exists a periodic $A' \subseteq \mathbb{Z}^2$ such that $F \oplus A' = \mathbb{Z}^2$. The recent refutation of the periodic tiling conjecture in high dimensions due to Greenfeld and Tao motivates finding different generalizations of Newman's theorem and of Bhattacharya's theorem that hold in arbitrary dimension $d$. In this paper, we formulate and prove such generalizations. We do so by studying the structure of joint co-tiles in $\mathbb{Z}^d$. Our generalization of Newman's theorem states that for any $d \ge 1$, any joint co-tile for $d$ independent tiles is periodic. For a $(d-1)$-tuple of finite subsets of $\mathbb{Z}^d$ that satisfy a certain technical condition that we call property $(\star)$, we prove that any joint co-tile decomposes into disjoint $(d-1)$-periodic sets. Consequently, we show that for a $(d-1)$-tuple of finite subsets of $\mathbb{Z}^d$ that satisfy property $(\star)$, the existence of a joint co-tile implies the existence of periodic joint co-tile. Conversely, we prove that if a finite subset $F$ in $\mathbb{Z}^d$ admits a periodic co-tile $A$, then there exist $(d-1)$ additional tiles that together with $F$ are independent and admit $A$ as a joint co-tile, so that the first $(d-2)$ of these tiles together with $F$ satisfy property $(\star)$. Combined, our results give a new necessary and sufficient condition for a subset of $\mathbb{Z}^d$ to tile periodically. We also discuss tilings and joint tilings in other countable abelian groups.