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Jan 7, 2014 at 22:49 comment added Pablo Shmerkin Thanks, I'm somewhat familiar with that paper but indeed while it gives a very good general bound if one wants $a$ to be not just smaller than $1$ but very small, it is certainly not $d$ for the $d-1$-dimensional sphere.
Jan 7, 2014 at 17:54 history answered Igor Rivin CC BY-SA 3.0