I've been thinking about proving some bounds for the OEIS sequence A319476:
$a(n)$ is the minimum number of distinct distances between $n$ non-attacking rooks on an $n \times n$ chessboard.
I'd like to find references to the minimum number of distinct distances between $n$ points in $\mathbb Z^2$. Call this minimum number of distances $a(n)$.
The minimum number of distinct distances in $\mathbb R^2$ is given by OEIS sequence A186704.
For example, $a(3) = 2$ because if there were only one distinct distance, this would mean there's a way to place the vertices of an equilateral triangle on $\mathbb Z^2$. We can, however, place an isosceles triangle like $(0, 0), (0, 1),$ and $(1, 0)$ on the grid, which has two distinct pairwise distances, namely $1$ and $\sqrt 2$.
Similarly, $a(4) = 2$ with vertices $(0, 0), (0, 1), (1, 0)$ and $(1, 1)$ and pairwise distances of $1$ and $\sqrt 2$.