Consider the $d$-dimensional $\ell_1$ ball $\mathbb B_d=\{x\in\mathbb R^d: \|x\|_1\leq 1\}$, where $\|x\|_1=\sum_{i=1}^d{|x_i|}$. I'm interested in the maximum size of the (finite) subset $S\subseteq\mathbb B_d$ such that for any two distinct elements $x,y\in S$, $\|x-y\|_{\infty}\geq \delta$, where $\|z\|_{\infty}=\max_{1\leq i\leq d}|z_i|$. $\delta$ is, of course, strictly between 0 and 1.

My main point of reference is Lemma 5.2 of the following note: https://www.stat.berkeley.edu/~wainwrig/nachdiplom/Chap5_Sep10_2015.pdf

If I understand it correctly, the (lower bound) side of Lemma 5.2 implies that

- If $\delta\lesssim 1/d$, then $\log |S|\asymp d\log(1/\delta)$ as $d\to\infty$, and this bound is almost tight;
- However, if $d\delta\to\infty$ it seems to me that the lower bound of $\log |S|$ becomes $\Omega(1)$ as $d\to\infty$, which clearly sounds loose to me.

My question is hence the following: **What is the asymptotic scaling of $\log |S|$ as $d\to\infty$ in the regime of $\delta\to 0$ and $d\delta\to\infty$? More concretely, if $\delta\asymp d^{-\alpha}$ for some $\alpha\in(0,1)$, how should $\log |S|$ scale with $d$ as $d\to\infty$? And furthermore, what if we consider the general $\ell_p$ ball $\{x\in\mathbb R^d: \|x\|_p\leq 1\}$ for $1\leq p<\infty$?**