Open access
The Number of Tiles of $\mathbb{Z}^d$
Mathematics
Abstract
For fixed $d\ge 1$, let $t_{n,d}$ be the number of subsets of$[n]^d$ that tile $\mathbb{Z}^d$ by translations. We prove that\[t_{n,d}=(3^{1/3})^{n^d\pm o(n^d)}.\]