Anti-concentration and the exact gap-Hamming problem

Anup Rao, Amir Yehudayoff

Research output: Contribution to journalArticlepeer-review


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 21.01n, 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: maxk\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 n2 of dimension 0.51n, then maxk Pr[\langle x, Y \rangle = k] \leq O(\sqrt{} ln(n)/n).

Original languageEnglish
Article number2
Pages (from-to)1071-1092
Number of pages22
JournalSIAM Journal on Discrete Mathematics
Issue number2
StatePublished - 2022


  • anticoncentration
  • communication complexity
  • probability

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


Dive into the research topics of 'Anti-concentration and the exact gap-Hamming problem'. Together they form a unique fingerprint.

Cite this