Some variants of the periodic tiling conjecture

The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\mathbb{1}_F * \mathbb{1}_A = 1$, there is also a periodic solution $\mathbb{1}_{A_{\mathrm{p}}}$. This conjecture is known to hold for some groups $G$ and fail for others. In this paper we establish three variants of the PTC. The first (due to Tim Austin) replaces the constant function $1$ on the right-hand side of the tiling equation by $0$, and the indicator functions $\mathbb{1}_F$ and $\mathbb{1}_A$ by bounded integer-valued functions. The second, which applies in $G=\mathbb{Z}^2$, replaces the right-hand side of the tiling equation by an integer-valued periodic function, and the functions $\mathbb{1}_F$ and $\mathbb{1}_A$ on the left-hand side by bounded integer-valued functions. The third (which is the most difficult to establish) is similar to the second, but retains the property of both $\mathbb{1}_A$ and $\mathbb{1}_{A_{\mathrm{p}}}$ being indicator functions; in particular, we establish the PTC for multi-tilings in $G=\mathbb{Z}^2$. As a result, we obtain the decidability of constant-level integer tilings in any finitely generated Abelian group $G$ and multi-tilings in $G=\mathbb{Z}^2$.