Is there a way to deduce the discrete results from the Gaussian result?
Lets look at the CLT:
CLT: If |a|2 = 1 and supi |ai| · d then
supx |P[εi ai xi · x] P[N · x]| · O(d)
Different formulation:
Let f : {-1,1}n ! R be a linear function: f(x) = ε ai xi and
|f|2 = 1.
I(f) · d.
Then supt |P[εi ai xi · t] P[εi ai Ni · t]| · O(d), where
Ni are i.i.d. Gaussians.