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Boosting uniformity in quasirandom groups: fast and simple

Harm Derksen, Chin Ho Lee, Emanuele Viola

TL;DR

It is shown that for any group <tex>$LI$</tex>, any distribution over <tex>$H^{m}$</tex> whose weight-k Fourier coefficients are small is close to a k-uniform distribution.

Abstract

We study the communication complexity of multiplying $k\times t$ elements from the group $H=\text{SL}(2,q)$ in the number-on-forehead model with $k$ parties. We prove a lower bound of $(t\log H)/c^{k}$. This is an exponential improvement over previous work, and matches the state-of-the-art in the area. Relatedly, we show that the convolution of $k^{c}$ independent copies of a 3-uniform distribution over $H^{m}$ is close to a $k$-uniform distribution. This is again an exponential improvement over previous work which needed $c^{k}$ copies. The proofs are remarkably simple; the results extend to other quasirandom groups. We also show that for any group $H$, any distribution over $H^{m}$ whose weight-$k$ Fourier coefficients are small is close to a $k$-uniform distribution. This generalizes previous work in the abelian setting, and the proof is simpler.

Boosting uniformity in quasirandom groups: fast and simple

TL;DR

It is shown that for any group <tex></tex>, any distribution over <tex></tex> whose weight-k Fourier coefficients are small is close to a k-uniform distribution.

Abstract

We study the communication complexity of multiplying elements from the group in the number-on-forehead model with parties. We prove a lower bound of . This is an exponential improvement over previous work, and matches the state-of-the-art in the area. Relatedly, we show that the convolution of independent copies of a 3-uniform distribution over is close to a -uniform distribution. This is again an exponential improvement over previous work which needed copies. The proofs are remarkably simple; the results extend to other quasirandom groups. We also show that for any group , any distribution over whose weight- Fourier coefficients are small is close to a -uniform distribution. This generalizes previous work in the abelian setting, and the proof is simpler.
Paper Structure (9 sections, 7 theorems, 26 equations)

This paper contains 9 sections, 7 theorems, 26 equations.

Table of Contents

  1. Introduction and our results
  2. $(\mathrm{\epsilon},k)$-uniformity vs. $k$-uniformity.
  3. Extensions.
  4. Preliminaries
  5. Matrices.
  6. Representation theory.
  7. Boosting uniformity
  8. Proof of \ref{['thm:cc-bound']}
  9. From $(\mathrm{\epsilon},k)$-uniform to $k$-uniform

Key Result

Theorem 1

Let $H=\mathrm{SL}(2,q)$. Let $P\colon H^{k\times t}\to[2]$ be a number-on-forehead communication protocol with $k$ parties and communication $b$ bits. For $g\in H$ denote by $p_{g}$ the probability that $P$ outputs $1$ over a uniform input $(a_{i,j})_{i\le k,j\le t}$ such that $\prod_{j=1}^{t}a_{1j

Theorems & Definitions (22)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Lemma 4: Schur's lemma, see MR964069, Page 11 or Lemma 2.3.3 in Wigderson-Barbados10
  • Definition 6: Gowers08
  • Theorem 7
  • Lemma 8: Lemma 3.3 in GowersV-cc-int-journal
  • proof
  • Lemma 9
  • Claim 10
  • ...and 12 more