Bounds on $d$ for tiling $\mathbb{Z}^d$ with subset of $\mathbb{Z}^n$?

Question

According to this remarkable paper by Gruslys, Leader and Tan, given any subset $T$ of $\mathbb{Z}^n$, $\exists d$ s.t. $T$ tiles $\mathbb{Z}^d$.

This immediately became one of my favourite combinatorial results of all time, but unfortunately the paper gives no bounds on $d$ in terms of $n$.

Do you know of any such bounds found in literature (or that you know of yourself) or, in the worst case scenario, any further work done on results of this kind?

Answer 1

In the concluding remarks, the paper does give the bound $d \le \lceil \exp(100(n \log k)^2)\rceil$ for a tile $T \subseteq [k]^n$ on the least dimension $d$ for which $T$ tiles $\mathbb Z^d$.

This paper is very recent and so it's unlikely that improved bounds have been found by anyone yet. This result (in more or less the same form as in the paper) also appears in Gruslys's 2018 PhD thesis, with just one additional remark on the bounds: it mentions a result of Metrebian (found here or here) that a symmetric punctured tile XXXXX.XXXXX of length $2k+1$ tiles $\mathbb Z^4$ for any $k$.

(Actually, a more recent result of Cambie shows that the symmetric punctured tile tiles $\mathbb Z^3$, which is best possible.)