Poly-logarithmic Independence Fools AC^0 Circuits

We prove that poly-sized AC
⁰
circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log
²
n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has later given a much simpler proof for Bazzipsilas theorem.