It is easy to construct a metric space $E_d$ such that all points of $E_d$ are at mutually integral distance and such that there is a map $\varphi$ from $E_d$ into the $d$-dimensional Euclidean space such that $\varphi$ preserves distances up to a bounded error and such that $\varphi(E_d)$ is uniformly dense (sufficiently large spheres centered at points of $\varphi(E_d)$ cover the Euclidean space): Take $\mathbb Z^d$ and define the distance between two distinct points $a,b\in\mathbb Z^d$ as the integer closest to $\lVert a-b\rVert+3$ (the $+3$ is probably not optimal).

*Is there a more natural metric space with the same properties? Is
there such a space which is optimal in some sense?*

`\parallel`

spaces poorly for norms; compare $\parallel a - b\parallel + 3$`\parallel a - b\parallel + 3`

to $\lVert a - b\rVert + 3$`\lVert a - b\rVert + 3`

. I have edited accordingly. $\endgroup$