The Number of Tiles of $\mathbb{Z}^d$

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
\[
t_{n,d}=(3^{1/3})^{n^d\pm o(n^d)}.
\]