Let $S^1$ be the unit circle of radius $1$.

For any $k\geq 1$, let the $k$-dimensional torus $T^k= \underbrace{S^1\times S^1\times\cdots\times S^1}_k$ be the $k$-fold self-Cartesian product of $S^1$.

Equip $T^k$ with the product Riemannian metric.

For any real number $r>0$ and any integer $n\geq 1$, consider the (ordered) disk configuration space

$$ F_r(T^k,n)=\Big\{(x_1,x_2,\ldots,x_n)\in T^k~~\Big |~~ d(x_i,x_j)\geq 2r ~{\rm ~for~any~}~1\leq i<j\leq n \Big \}. $$

Fix $r$ and $k$.

If $n$ is large enough, then $F_r(T^k,n)=\emptyset$.

Denote the largest $n$ such that $F_r(T^k,n)\neq \emptyset$ by $N(k,r)$.

**Question 1.** *For any integer $k\geq 1$ and any real number $r>0$, how to compute $N(k,r)$?*

**Question 2.** *For any real number $r>0$, how to compute the limit $\lim_{k\to\infty }N(k,r)^{\frac{1}{k}}$?*

Are there any references?

Thank you very much.