We work in $\mathbb{R}^n$ (or even just $n=2$ is interesting). Let $k$ be a positive integer and $N(k)$ be the maximum number of open balls radius $1/k$ which can be (disjointly) arranged in the unit ball. I want to know what the asymptotic order of $N(k)$ is.
For example, the volume of the balls is proportional to $N(k)k^{-n}$ so certainly $N(k)\le k^{n}$. Is this the correct asymptotic order?
Intuitively, I'd guess that we can cover $1-\epsilon$ proportion of the unit ball for any $\epsilon>0$ and $k$ suitably large, so I think $N(k) \sim k^n$.
EDIT: It turns out, this is almost identical to a general version of the (now proven) Kepler Conjecture, which gives the bound $N(k)\le 0.741 k^{-3}$ for $n=3$. I still think we should be able to show $N(k)=\Omega(k^n)$.