Israel Journal of Mathematics

, Volume 77, Issue 1, pp 55–64

The influence of variables in product spaces

  • Jean Bourgain
  • Jeff Kahn
  • Gil Kalai
  • Yitzhak Katznelson
  • Nathan Linial
Article

DOI: 10.1007/BF02808010

Cite this article as:
Bourgain, J., Kahn, J., Kalai, G. et al. Israel J. Math. (1992) 77: 55. doi:10.1007/BF02808010

Abstract

LetX be a probability space and letf: Xn → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byIf(k), as follows: Foru=(u1,u2,…,un−1) ∈Xn−1 consider the setlk(u)={(u1,u2,...,uk−1,t,uk,…,un−1):tX}.\(I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)).\)

More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byIf(S), be the probability that assigning values to the variables not inS at random, the value off is undetermined.

Theorem 1:There is an absolute constant c1so that for every function f: Xn → {0, 1},with Pr(f−1(1))=p≤1/2,there is a variable k so that\(I_f (k) \geqslant c_1 p\frac{{\log n}}{n}.\)

Theorem 2:For every f: Xn → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c2(ε)n/logn so that If (S)≥1−ε.

These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}.

Copyright information

© Hebrew University 1992

Authors and Affiliations

  • Jean Bourgain
    • 1
  • Jeff Kahn
    • 2
  • Gil Kalai
    • 3
    • 4
  • Yitzhak Katznelson
    • 5
  • Nathan Linial
    • 3
  1. 1.Département de MathématiquesIHESBures-sur-YvetteFrance
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA
  3. 3.Landau Center for Research in Mathematical Analysis Institute of Mathematics and Computer ScienceThe Hebrew University of JerusalemJerusalemIsrael
  4. 4.IBM Almaden Research CenterIsrael
  5. 5.Department of MathematicsStanford UniversityStanfordUSA

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