Israel Journal of Mathematics

, Volume 214, Issue 1, pp 167–192 | Cite as

On the sum of the L 1 influences of bounded functions

  • Yuval Filmus
  • Hamed Hatami
  • Nathan Keller
  • Noam Lifshitz
Article

Abstract

Let f: {-1, 1} n → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L 1 influence of f can also be bounded as a function of d. Bačkurs and Bavarian answered this question in the affirmative, providing a bound of O(d 3) for general functions and O(d 2) for homogeneous functions. We improve on their results by providing a bound of d 2 for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.

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Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  • Yuval Filmus
    • 1
  • Hamed Hatami
    • 2
  • Nathan Keller
    • 3
  • Noam Lifshitz
    • 3
  1. 1.Computer Science DepartmentTechnion–Israel Institute of TechnologyHaifaIsrael
  2. 2.School of computer science and the Department of mathematics and statisticsMcGill universityMontrealCanada
  3. 3.Department of MathematicsBar Ilan UniversityRamat GanIsrael

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