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

Periodicity of joint co-tiles in $\mathbb{Z}^d$, Discrete Analysis 2024:13, 32 pp.

A finite subset, or tile, $F$ of an abelian group $G$ is said to tile $G$ by translations if there there is some complementary set, or co-tile, $A$ of $G$ such that the translates $a+F$, $a \in A$ of the tile form a partition of $G$. For instance, the sets $\{0,1\}$, $\{0,2\}$, and $\{0,1,2\}$ each tile the integers ${\mathbb Z}$ by translations, but $\{0,1,3\}$ does not. In some cases, the tiling is fully periodic, which means that the set of periods of the co-tile $A$ that is, elements $\gamma$ of the group such that $\gamma + A = A$, form a finite index subgroup of $G$; on other cases, the tiling may only have a limited amount of periodicity (the group of periods may have a lower rank than $G$), or no periodicity at all. For instance, in the lattice ${\mathbb Z}^2$, the square $\{0,1\}^2$ admits tilings that are fully periodic, as well as tilings that are periodic in only one direction.

The periodic tiling conjecture asserts that if a set $F$ can tile an abelian group $G$ at all, then it must also admit periodic tilings. This was shown by Newman in the one-dimensional case $G = {\mathbb Z}$ (in fact he observed that all tilings are periodic in this case), and by Bhattacharya in the two-dimensional case $G = {\mathbb Z}^2$. In the latter case, non-periodic tilings can exist, but Greenfeld and Tao showed here that all tilings are “weakly periodic”, which means that the co-tile $A$ is a disjoint union of finitely many sets, each of which is periodic in at least one direction. On the other hand, another result of Greenfeld and Tao shows that if $G = {\mathbb Z}^d$ for large enough $d$, then the periodic tiling conjecture is false: there exist tiles that can tile ${\mathbb Z}^d$, but not in a way which is fully periodic.

However, it seems that some remnant of the periodic tiling conjecture still persists in these settings, in that at least certain types of tilings are still forced to obey some periodicity properties. Define a set $F_1,\dots,F_k$ of tiles in $\mathbb Z^d$ to be independent if each $F_i$ contains 0 and if $v_1,\dots,v_k$ are non-zero elements of $F_1,\dots,F_k$, respectively, then $v_1,\dots,v_k$ are linearly independent. The authors here show that if a set $A$ in ${\mathbb Z}^d$ is a co-tile for $d$ independent tiles, then it must be fully periodic (generalizing the result of Newman), and if it is a co-tile for $d-1$ independent tiles, then it is the finite union of sets that are each periodic in $d-1$ different directions (generalizing the first result of Greenfeld and Tao), and one can replace that co-tile with a fully periodic set (generalizing the result of Bhattacharya). These results also have converses: for instance, if a tile $F$ admits a fully periodic co-tile $A$, then this co-tile $A$ is also a co-tile for $d-1$ other independent tiles $F$. Thus this paper gives a link between the periodicity properties of co-tiles, and the number of independent tiles that they are co-tiles for.