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

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}.

## 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