## Abstract

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 ^{n}_{2} of dimension 0.51n, then max_{k} Pr[\langle x, Y \rangle = k] \leq O(\sqrt{} ln(n)/n).

Original language | English |
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Article number | 2 |

Pages (from-to) | 1071-1092 |

Number of pages | 22 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - 2022 |

## Keywords

- anticoncentration
- communication complexity
- probability

## All Science Journal Classification (ASJC) codes

- Mathematics(all)