Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound

We give an explicit function $h:\{0,1\}^n \to \{0,1\}$ such that every de Morgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. Our technical contributions include a theorem that shows that the “expected shrinkage” result of H\aastad [SIAM J. Comput., 27 (1998), pp. 48--64] actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), using ideas of Impagliazzo, Meka, and Zuckerman [Proceedings of FOCS, 2012, pp. 111--119].