This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\mathbb C^n$, I want to pick a large family of vectors $(u\_i)\_{i=1}^k$ which is almost orthogonal in the sense that $|(u\_i|u\_j)| < \epsilon$ when $i\not=j$. I guess I'm interested in how the biggest choice of $k$ grows with $n$ and $\epsilon$.


For example, we can let $\{u\_1,\cdots,u\_n\}$ be the usual basis, and then choose $u\_{n+1} = (1,1,\cdots,1)/n^{1/2}$, which works if $n^{-1/2} < \epsilon$. Then you can let $u\_{n+2} = (1,\cdots,1,-1,\cdots,-1)/n^{1/2}$ and so forth, but it's not clear to me how far you can go.



Almost orthogonal vectors
Matt, to get $k$ points, you only need $n\ge C \epsilon^{-2} \log k$. See <http://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma> or Google "Johnson-Lindenstrauss lemma".


