Author here. I know that treating the L^2 ball as somehow equivalent to binary vectors is invalid - I even say so in the post. If it helps, I explicitly point o

Author here. I know that treating the L^2 ball as somehow equivalent to binary vectors is invalid - I even say so in the post. If it helps, I explicitly point out the place where the real work for the binary vector would have to go. The basic idea is I think is to use the central limit theorem on dot(u,d) and dot(p,d) - but unfortunately the CLT does not show that these two dot products are independent.