A strong direct product theorem for corruption and the multiparty communication complexity of disjointness

We prove that two-party randomized communication complexity satisfies a strong direct product property, so long as the communication lower bound is proved by a "corruption" or "one-sided discrepancy" method over a rectangular distribution. We use this to prove new n ^Ω(1) lower bounds for 3-player number-on-the-forehead protocols in which the first player speaks once and then the other two players proceed arbitrarily. Using other techniques, we also establish an Ω(n ^1/(k-1)/(k - 1)) lower bound for k-player randomized number-on-the-forehead protocols for the disjointness function in which all messages are broadcast simultaneously. A simple corollary of this is that general randomized number-on-the-forehead protocols require Ω(log n/(k - 1)) bits of communication to compute the disjointness function.