4 problems

Let $V$ be a finite tile in $\mathbb{Z}^n$. A tiling by translations is fully periodic if its translation set has a finite-index period subgroup. Lagarias--Wang conjecture. If $V$…

Periodic tiling conjecture. Any $E$ which tiles ${\mathbb R}^n$ by translations must also admit a periodic tiling.

Weakly periodic tiling conjecture. There exists a weakly periodic $F$-tiling of $\mathbb Z^d$; more precisely, there exists an $F$-tiling $T\subseteq\mathbb Z^d$ admitting such a f…

Periodic tiling conjecture. If there is an $F$-tiling of $\mathbb Z^n$ for some cluster $F$, then there is a fully periodic $F$-tiling.