ANTICONCENTRATION 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 {+/- 1}(n) with vertical bar A vertical bar .vertical bar B vertical bar 2(1.01n), and X is an element of A and Y is an element of B are sampled independently and uniformly, then the inner product < X, Y > takes on any fixed value with probability at most O(1/root n). In fact, we prove the following stronger "smoothness" "statement: max(k) vertical bar Pr[< X, Y > = k] Pr[< X, Y >= k + 4] <= 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 is an element of Z(n) has no zero coordinates, and B subset of{+/- 1}(n) corresponds to a subspace of F-2(n) of dimension 0.51n, then max(k) Pr[< x, Y > = k <= O(root ln(n)/n).