Anti-concentration in most directions

We prove anti-concentration bounds for the inner product of two independent random vectors. For example, we show that if $A,B$ are subsets of the cube $\{\pm 1\}^n$ with $|A| \cdot |B| \geq 2^{1.01 n}$, 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 $O(\tfrac{1}{\sqrt{n}})$. Extending Halász work, we prove stronger bounds when the choices for $x$ are unstructured. We also describe applications to communication complexity, randomness extraction and additive combinatorics.