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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. What I want is that for any vector $z\in\{0,1\}^n$, the projection of $z$ onto $L$ would be of length at most $\|z\|/\log\log n$ (say). What properties of $L$ can ensure this?

I do believe this is a most real and very reasonable question. I will be happy to provide more explanations, if needed.

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?

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!

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. What I want is that for any vector $z\in\{0,1\}^n$, the projection of $z$ onto $L$ would be of length at most $\|z\|/\log\log n$ (say). What properties of $L$ can ensure this?

I do believe this is a most real and very reasonable question. I will be happy to provide more explanations, if needed.