An Optimal "It Ain't Over Till It's Over" Theorem

We study the probability of Boolean functions with small max influence to become constant under random restrictions. Let $f$ be a Boolean function such that the variance of $f$ is $\Omega(1)$ and all its individual influences are bounded by $\tau$. We show that when restricting all but a $\rho=\tilde{\Omega}((\log(1/\tau))^{-1})$ fraction of the coordinates, the restricted function remains nonconstant with overwhelming probability. This bound is essentially optimal, as witnessed by the tribes function $\mathrm{TRIBES}=\mathrm{AND}_{n/C\log n}\circ\mathrm{OR}_{C\log n}$. We extend it to an anti-concentration result, showing that the restricted function has nontrivial variance with probability $1-o(1)$. This gives a sharp version of the "it ain't over till it's over" theorem due to Mossel, O'Donnell, and Oleszkiewicz. Our proof is discrete, and avoids the use of the invariance principle. We also show two consequences of our above result: (i) As a corollary, we prove that for a uniformly random input $x$, the block sensitivity of $f$ at $x$ is $\tilde{\Omega}(\log(1/\tau))$ with probability $1-o(1)$. This should be compared with the implication of Kahn, Kalai, and Linial's result, which implies that the average block sensitivity of $f$ is $\Omega(\log(1/\tau))$. (ii) Combining our proof with a well-known result due to O'Donnell, Saks, Schramm, and Servedio, one can also conclude that: Restricting all but a $\rho=\tilde\Omega(1/\sqrt{\log (1/\tau) })$ fraction of the coordinates of a monotone function $f$, then the restricted function has decision tree complexity $\Omega(\tau^{-\Theta(\rho)})$ with probability $\Omega(1)$.