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Let $k$, $m$ and $n$ be positive integers. Let $r$ be a positive real number.

The $n$-th ordered $r$-disk configuration space on the Euclidean space $\mathbb{R}^{mk}$ is

$$ F_r(\mathbb{R}^{mk}, n)=\Big\{(x_1,x_2,\ldots,x_n)\in \underbrace{ \mathbb{R}^{mk}\times \mathbb{R}^{mk}\times\cdots\times\mathbb{R}^{mk}}_n~~\Big |~~||x_i||_{\mathbb{R}^{mk}}\leq 1 {\rm~~for~any~~}1\leq i\leq n {\rm~~and}~~ d_{\mathbb{R}^{mk}}(x_i,x_j)\geq 2r {\rm~~for~any~~} 1\leq i<j\leq n\Big\}. $$

Let $N(r,m,k)$ be the largest integer $n$ such that $F_r(\mathbb{R}^{mk}, n)\neq \emptyset$.

Fix $r$ and $m$.

Question. Whether is it true or not that $N(r,m,k_1)^{1/k_1}\leq N(r,m,k_2)^{1/k_2}$ for any $k_1>k_2$?

Question. If the limit $\lim_{k\to\infty} N(r,m,k)^{1/k}$ exists, how to compute or estimate the limit $\lim_{k\to\infty} N(r,m,k)^{1/k}$?

Question. Suppose in addition, $r$ is sufficiently small. For example, $r=0.0001$, $r=0.001$. Are there any results or methods for $\lim_{k\to\infty} N(r,m,k)^{1/k}$?

Are there any such references? Thank you very much.

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    $\begingroup$ To be clear, this is value is only a function of the dimension $d=mk$, correct? I find the separation of variables somewhat strange to think about in this context versus a function $N(r,d)$. $\endgroup$
    – RavenclawPrefect
    Commented Dec 15, 2021 at 12:53
  • $\begingroup$ Yes, Prof. @RavenclawPrefect $\endgroup$
    – Shiquan Ren
    Commented Dec 16, 2021 at 1:33

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