Let $K_n$ be the sets of vectors $x \in \mathbb{Z}^d $ with each coordinates $x_i$ between $1$ and $n$. For any subset $A$ of $K_n$, let $S(A)$ be the set of points $x \in K_n$ which are on some line containing at least two points of $A$ (in other words, $S(A)$ is the union of the lines passing through - at least - two points of $A$).

Such a set $A$ is said to *generate* $K_n$ if $S(A) = K_n$. Now let $r_d(n)$ be the smallest size of a generating subset of $K_n$.

Question : What are the best known bounds on $r_d(n)$ ? (The first non trivial case is $d=2$)

This problem may be "well-known" ; I'm almost sure this question has already been studied, but I didn't find any reference, and Google gives nothing.

The trivial bound is $r_d(n) \gg_d n^{\frac{d}{2}- \frac{1}{2}} $ : taking a generating subset of size $r_d(n)$, there are at most $O(r_d(n)^2)$ lines to consider, each one intersecting $K_n$ in at most $n$ points, so that $|K_n| \ll r_d(n)^2 \times n$.

A refinement of this argument (a typical line contains much less than $n$ points of $K_n$) gives a lower bound $r_d(n) \gg_d n^{\frac{d}{2}- \frac{1}{4} - \frac{1}{4(2d-1)} } $.