Let $0<\delta<1$. We say that $\{e_{1},\cdots,e_{m}\}\subset\mathbb{S}^{n-1}$ is a $\delta$-separated subset of $\mathbb{S}^{n-1}$ if $|e_{j}-e_{k}|\geq\delta$ for $j\neq k$. It is maximal if in addition for every $e\in\mathbb{S}^{n-1}$ there is some $k$ for which $|e-e_{k}|<\delta$.
Given a maximal $\delta$-separated subset $\{e_{1},\cdots,e_{m}\}\subset\mathbb{S}^{n-1}$ I would like to prove the following: for any fixed $k$,
\begin{equation*} \sum_{l=1 \\ l\neq k}^{m}\frac{\delta}{|e_{k}-e_{l}|} \lesssim \log(1/\delta). \end{equation*}
By dividing the sphere into sectors with points whose distance to $e_{k}$ is between $i\delta$ and $(i+1)\delta$, if one proves that the number of $e_{l}$ in each sector is uniformly bounded (by a constant not depending on $\delta$), then one can get this bound by summing over the $e_{l}$ with $|e_{k}-e_{l}|\leq 1$, having yet to account for the contribution of those such that $|e_{k}-e_{l}|> 1$. Is this right or is there a shortcut to prove this bound? Thanks!