Tight Bounds on the Fourier Spectrum of AC0

We show that AC0 circuits on n variables with depth d and sizem have at most 2−Ω(k/ logd−1 m) of their Fourier mass at level k or above. Our proof builds on a previous result by Håstad (SICOMP, 2014) who proved this bound for the special case k = n. Our result improves the seminal result of Linial, Mansour and Nisan (JACM, 1993) and is tight up to the constants hidden in the Ω notation. As an application, we improve Braverman’s celebrated result (JACM, 2010). Braverman showed that any r(m, d, ε)-wise independent distribution ε-fools AC0 circuits of size m and depth d, for r(m, d, ε) = O(log(m/ε))2d 2+7d+3. Our improved bounds on the Fourier tails of AC0 circuits allows us to improve this estimate to r(m, d, ε) = O(log(m/ε))3d+3. In contrast, an example by Mansour (appearing in Luby and Velickovic’s paper – Algorithmica, 1996) shows that there is a logd−1(m) · log(1/ε)-wise independent distribution that does not ε-fool AC0 circuits of size m and depth d. Hence, our result is tight up to the factor 3 in the exponent. 1998 ACM Subject Classification F.1.3 [Computation by Abstract Devices] Complexity Measures and Classes