Anticoncentration and the Exact Gap-Hamming Problem

We prove anti-concentration bounds for the inner product of two independent random vectors, and use these to lower in communication complexity. show that if $A,B$ are subsets cube $\{\pm 1\}^n$ with $|A| \cdot |B| \geq 2^{1.01 n}$, $X \in A$ $Y B$ sampled independently uniformly, then $\langle X,Y \rangle$ takes on any fixed value probability at most $O(1/\sqrt{n})$. In fact, we following stronger "smoothness" statement: $$ \max_{k } \big| \Pr[\langle \rangle = k] - k+4]\big| \leq O(1/n).$$ results exact gap-hamming problem requires linear communication, resolving an open also conclude structured distributions low entropy. If $x \mathcal{Z}^n$ has no zero coordinates, $B \subseteq \{\pm corresponds a subspace $\mathcal{F}_2^n$ dimension $0.51n$, $\max_k x,Y O(\sqrt{\ln (n)/n})$.