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Fourier Conjectures, Correlation Bounds, and Majority

Author Emanuele Viola

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Emanuele Viola
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA

Funding

  • Viola, Emanuele: Supported by NSF CCF award 1813930.

Acknowledgements

This paper includes the results in [Emanuele Viola, 2019]. I am grateful to Chin Ho Lee for pointing out the work [Eshan Chattopadhyay et al., 2020] to me, and to an anonymous reviewer for suggesting the use of hypercontractivity to bound 𝔼|g_{k}(x)| in the proof of Theorem 1 (alternatively one can reason along the lines of the proof of Theorem 4). A preliminary version of this paper had Theorem 7 only for d ≥ Ω(n^{1/3}), and the degree bound was O(d√{log n}). Jarosław Błasiok pointed out to us how to improve the proof to obtain Theorem 7. The proof in the preliminary version was similar, but rather than performing a case analysis, detected the two cases explicitly with an auxiliary polynomial, which led to d ≥ Ω(n^{1/3}). It also used the polynomials for {Maj} with polynomially-small error, as opposed to constant, which led to the extra √{log n} factor. Following these ideas, we also improved the results on the coin problem and h₂. We are very grateful to Jarosław Błasiok for letting us include the improved results!

Cite AsGet BibTex

Emanuele Viola. Fourier Conjectures, Correlation Bounds, and Majority. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 111:1-111:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.111

Abstract

Recently several conjectures were made regarding the Fourier spectrum of low-degree polynomials. We show that these conjectures imply new correlation bounds for functions related to Majority. Then we prove several new results on correlation bounds which aim to, but don't, resolve the conjectures. In particular, we prove several new results on Majority which are of independent interest and complement Smolensky’s classic result.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Fourier analysis
  • polynomials
  • Majority
  • correlation
  • lower bound
  • conjectures

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