Upper and Lower Bounds on the Complexity of Unit-Distance Graph Realization in $\mathbb{Z}^d$

It is known that the problem of recognizing the existence of a unit-distance realization in $\mathbb{Z}^d$ for a graph $G$ is NP-complete. Here, assuming ETH, we give a sub-exponential lower bound of $2^{Ω(\sqrt{n})}$ for this problem in $\mathbb{Z}^2$ and $2^{Ω(n^{2/3})}$ in $\mathbb{Z}^3$, and giv