The conjecture on achievable burning densities of growing grids in dimensions greater than two

Let $f: \mathbb{N} \to \mathbb{N}$ be a strictly increasing function, let

$$G_n=[-f(n),f(n)]^d$$

for some $d>2$, and let $\mathcal G(f)=(G_n,n\geq 1)$. Write $P(\mathcal G(f))$ for the set of achievable burning densities. Growing-grid burning density conjecture. The following assertions hold:

1. If $f(n)=\lceil cn\rceil$ for some $c\geq 1$, then

$$P(\mathcal G(f))=[1/(2c^d),1].$$

2. If $f(n)=\omega(n)$ and $f(n)=o(n^{(d+1)/d})$, then

$$P(\mathcal G(f))=[0,1].$$

3. If $f(n)=\lceil cn^{(d+1)/d}\rceil$ for some $c>0$, then

$$P(\mathcal G(f))=[0,\phi(c,d)]$$

for some constant $\phi(c,d)$ depending on $c$ and $d$. 4. If $f(n)=\omega(n^{(d+1)/d})$, then

$$P(\mathcal G(f))=\{0\}.$$

This conjecture extends the paper’s two-dimensional results to growing $d$-dimensional grids. The supplied text presents all four cases as a proposed natural extension and gives no evidence that they have been proved or disproved.