I have some (rather exotic) subspace $L<R^n$, and I want to show that every non-zero vector in $\{0,1\}^n$ has a relatively small projection onto $L$. What general results and tools can be helpful? Anything from geometry of numbers?

Any suggestions appreciated!

Upon looking at the responses, some explanations may be in order. Let's say that a subspace $L<R^n$ is *oblique* if for any vector $z\in\{0,1\}^n$, the projection of $z$ onto $L$ is of length at most $\|z\|/\log\log n$ (say). What properties of a subspace can ensure that it is oblique? Can any general "obliqueness criteria" be given?

notaligned with the coordinate planes! – Seva Apr 5 '11 at 6:53