Version 2 changes: ----------------- (All lemma/theorem/section/etc. numbers are according to the version 1 numbering) -- Fixed various typos + other minor changes (2/18/2010) -- Removed Lemma G.5 from the Appendix (it was wrong). The error is that when you "choose the plane that X and Y lie in before choosing anything else", what you're left with after conditioning is no longer Gaussian (see "On the maximal perimeter of a convex set in R^n with respect to a Gaussian measure" by Fedja Nazarov and "The Reverse Isoperimetric Problem for Gaussian Measure" by Keith Ball for more on anticoncentration around the boundary of a convex set under Gaussian measure). The net effect is that Theorem G.6 in the Appendix reduces the m^6 dependence of Theorem 8.1 to m^4, not m^2. (2/18/2010) ----------------- -- An early version of this paper advertised a "new" spectral moment bound for quadratic forms. This bound turns out to not be new, and is known as the "Hanson-Wright inequality" (thanks to Assaf Naor for a relevant reference). The current version correctly attributes the bound, though we've left our proof in there since it seems to be different from the other proofs we've found of the Hanson-Wright inequality. (10/13/2010)