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Searching for Regularity in Bounded Functions

Authors Siddharth Iyer, Michael Whitmeyer

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  • Mathematics of computing
Keywords
  • regularity
  • bounded function
  • Boolean function
  • Fourier analysis

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Abstract

Given a function f on F₂ⁿ, we study the following problem. What is the largest affine subspace 𝒰 such that when restricted to 𝒰, all the non-trivial Fourier coefficients of f are very small? 
For the natural class of bounded Fourier degree d functions f: F₂ⁿ → [-1,1], we show that there exists an affine subspace of dimension at least Ω(n^{1/d!} k^{-2}), wherein all of f’s nontrivial Fourier coefficients become smaller than 2^{-k}. To complement this result, we show the existence of degree d functions with coefficients larger than 2^{-d log n} when restricted to any affine subspace of dimension larger than Ω(d n^{1/(d-1)}). In addition, we give explicit examples of functions with analogous but weaker properties.
Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of F₂ⁿ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.

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Siddharth Iyer and Michael Whitmeyer. Searching for Regularity in Bounded Functions. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.83

Author Details

Siddharth Iyer
  • University of Washington CSE, Seattle, WA, USA
Michael Whitmeyer
  • University of Washington CSE, Seattle, WA, USA

Funding

  • Iyer, Siddharth: Research supported by NSF grant CCF-2131899.
  • Whitmeyer, Michael: Research supported by NSF grant CCF-2006359.

Acknowledgements

We thank Anup Rao for posing the question that launched this project and for his invaluable advice and feedback. We are also grateful Paul Beame for his extremely helpful advice, discussions, and feedback. Finally, we thank Sandy Kaplan for detailed feedback on this writeup.

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