Consider the distribution given by taking the top singular vector of a matrix whose entries are equally likely to be +1 or -1. This is not well-defined for matrices where the subspace achieving the top singular value is >1-dimensional, but we can ignore this case since it occurs with negligible probability.
Can we say anything obvious about how this distribution compares to the uniform distribution on unit norm vectors (that is, the distribution on the top singular vector of an iid Gaussian matrix); say, that the support of the distribution forms some $\varepsilon$-net of the general space, or that no matter what direction I choose, the probability that I pull a singular vector that is close to that direction is about uniform?
Essentially I'm looking for some discrete analogue of rotational invariance, and I want to determine whether this distribution satisfies such a property.
Sorry if this is well-known, but I would appreciate being pointed to the relevant literature.