Anti-concentration in most directions

TL;DR

This paper establishes anti-concentration bounds for the inner product of independent random vectors from large subsets of the hypercube, with implications for communication complexity, randomness extraction, and additive combinatorics.

Contribution

It provides new anti-concentration bounds for inner products in high dimensions, extending previous work and applying to various areas in theoretical computer science.

Findings

Inner product takes any fixed value with probability at most O(1/√n) for large subsets.

Stronger bounds are achieved for unstructured choices of vectors.

Applications include improved results in communication complexity and additive combinatorics.

Abstract

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 Hal\'asz work, we prove stronger bounds when the choices for $x$ are unstructured. We also describe applications to communication complexity, randomness extraction and additive combinatorics.