On the number of tiles in $[n]^d$

Abstract

Let $\mathcal{T}(n,d)$ denote the set of subsets of $[n]^d$ that tile $\mathbb Z^d$ by translations, and write $T(n,d)=|\mathcal{T}(n,d)|$. We show that almost every tile in $\mathcal{T}(n,1)$ tiles $\mathbb{Z}$ by an arithmetic progression, thus determining $T(n,1)$ asymptotically. In higher dimensions, we prove that $T(n,d)=O_d\!\left(n^{C_d}3^{n^d/3}\right)$, confirming a conjecture of Benjamini, Kozma and Tzalik. In the other direction, we prove that $T(n,d)\geq (1-o(1))n^{\binom d2}3^{n^d/3}$ when $3\mid n$. This shows one cannot take $C_d=d-1$ for $d\geq 3$, answering a question due to the same authors negatively. Finally, we also obtain nearly tight bounds in dimension two. Our upper bounds rely on a new observation realising tiles as maximal independent sets in truncated Cayley graphs.