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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.

Boby
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    Unless $R/r$ is small, say between $1$ and $10,$ you are not going to do much better than the best lattice packing for that dimension. See SPLAG, Sphere Packing, Lattices, and Groups, by John H. Conway and Neil Sloane – Will Jagy Nov 11 '16 at 19:16
  • @WillJagy So, I tried to check that book. I can not find bound on $N$ in terms of $R,r,n$. Ideally, I want to look at the case when $R>>r$. – Boby Nov 11 '16 at 21:46
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    Minkowski-Hlawka's theorem tell us in $n$-dimesion, there is a lattice packing with packing density $\Delta \ge \frac{\zeta(n)}{2^{n-1}}$. There are attempts to improve this bound. The best I know is $\frac{(n-1)\zeta(n)}{2^{n-1}}$. So for $R \gg r$ and large $n$, we only have something like $N \ge O\left(\frac{n}{2^n}\left(\frac{R}{r}\right)^n\right)$. – achille hui Nov 12 '16 at 07:24
  • @achillehui Will be grateful if you could put this as an answer and add more details. – Boby Nov 12 '16 at 14:23
  • @TakahiroWaki I went through your link. Can you put to the theorem I should use? – Boby Nov 21 '16 at 13:24
  • @Boby Henry Cohn, Abhinav Kumar, Noam Elkies are famous in this research, and Elkies is in this community. How about do you ask him? – Takahiro Waki May 02 '17 at 09:49

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