Let $B$ be a closed ball in $\mathbb R^d$ of radius $R$ and let $N=N_R(r)$ denote the maximal cardinality of the $r$-separated sets (meaning any two points in this set have distance at least $r$) that can be contained in $B$.

If $r>2R$ then $N_R(r)=1$.

If $r\le 2R$, I expect $N$ can be bounded by something like $C(\frac{2R}{r})^{d+\delta}$ where $C,\delta$ are absolute constants (I may be wrong here. Let me know the correct form).

I wonder what are naive/classical/latest results on this estimation.