It is well-known that the size of an $\varepsilon$-net in $\mathbb{R}^n$ is upper bounded by $(1+2/\varepsilon)^n$. What if I only wanted to counted the number of elements within this net which is $s$-sparse, i.e., has support only on $s$ coordinates?
One trivial way to do it would be, first pick the $s$ coordinates and there are $\binom{n}{s}$ ways to do it and then fit in an $\varepsilon$-ball inside these $s$ coordinates, to get $\binom{n}{s}\cdot (1/\varepsilon)^s$, but i dont know if this is correct: suppose we had a sparsity pattern $P_1,P_2$ with centers $a,b$ respectively, then why should two vectors $x,y$ lying inside $P_1$ and $P_2$ be $\varepsilon$-far? By triangle inequality we get that $d(x,y)\geq d(a,b)-2\varepsilon$, but i dont know why this is large.
Any references would be appreciated.