This question has probably been asked before but when I searched the site I could not find the answer.
Suppose we have and $n$-dimensional ball with radius $R$. How many, smaller $n$-dimensional ball with radius $r$ can we fit in this ball. Let this number be denoted by $N$.
I am aware that this is still an open question in math. But can we give some lower and upper bounds on $N$?
For example,
\begin{align}
N \le \frac{{\rm Vol}(R)}{{\rm Vol}(r)}= \left(\frac{R}{r} \right)^n,
\end{align}
My questions is: Are there any non-asymptotic lower bounds on $N$ in terms of $R,r,n$?
If this question has been answered in this site before. Please direct me to it. There was and answer in the comments that we can have an asymptotic bound by Minkowski–Hlawka theorem. However, I would like to see more explanations on how it relates.
For the bounty, I would really like a precise argument possible with some references.