Anti-concentration and the exact gap-Hamming problem

We prove anticoncentration bounds for the inner product of two independent random vectors and use these bounds to prove lower bounds in communication complexity. We show that if A, B are subsets of the cube \{ \pm 1\} ⁿ with | A| \cdot | B| \geq 2^1.01n, and X \in A and Y \in B are sampled independently and uniformly, then the inner product \langle X, Y \rangle takes on any fixed value with probability at most O(1/\surd n). In fact, we prove the following stronger ``smoothness"" statement: max_k^{\bigm| \bigm|} Pr[\langle X, Y \rangle = k] - Pr[\langle X, Y \rangle = k + 4]^{\bigm| \bigm|} \leq O(1/n). We use these results to prove that the exact gap-hamming problem requires linear communication, resolving an open problem in communication complexity. We also conclude anticoncentration for structured distributions with low entropy. If x \in \BbbZ ⁿ has no zero coordinates, and B \subseteq \{ \pm 1\} ⁿ corresponds to a subspace of \BbbF ⁿ₂ of dimension 0.51n, then max_k Pr[\langle x, Y \rangle = k] \leq O(\sqrt{} ln(n)/n).