Abstract
LetX be a probability space and letf: X n → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI f (k), as follows: Foru=(u 1,u 2,…,u n−1) ∈X n−1 consider the setl k (u)={(u 1,u 2,...,u k−1,t,u k ,…,u n−1):t ∈X}.\(I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)).\)
More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI f (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 c 1 so that for every function f: X n → {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: X n → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c 2(ε)n/logn so that I f (S)≥1−ε.
These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Alon,On the density of sets of vectors, Discrete Math.46 (1983), 199–202.
W. Beckner,Inequalities in Fourier analysis, Annals of Math.102 (1975), 159–182.
M. Ben-Or and N. Linial,Collective coin flipping, inRandomness and Computation (S. Micali ed.), Academic Press, New York, 1990, pp. 91–115. Earlier version:Collective coin flipping, robust voting games, and minima of Banzhaf value, Proc. 26th IEEE Symp. on the Foundation of Computer Science, 1985, 408–416.
B. Bollobas,Combinatorics, Cambridge University Press, Cambridge, 1986.
B. Chor and M. Gereb-Graus,On the influence of single participant in coin flipping schemes, Siam J. Discrete Math.1 (1988), 411–415.
P. Frankl,The shifting technique in extremal set theory, inSurveys in Combinatorics (C. W. Whitehead ed.), Cambridge University Press, Cambridge, 1987, 81–110.
J. Kahn, G. Kalai and L. Linial,The influence of variables on Boolean functions, Proc. 29th Ann. Symp. on Foundations of Comp. Sci., Computer Society Press, 1988, pp. 68–80.
J. Kahn, G. Kalai and L. Linial,Collective coin flipping, the influence of variables on Boolean functions and harmonic analysis, to appear.
L. Loomis and H. Whitney,An inequality related to the isoperimetric inequality, Bull. Am. Math. Soc.55 (1949), 961–962.
Author information
Authors and Affiliations
Additional information
Work supported in part by grants from the Binational Israel-US Science Foundation and the Israeli Academy of Science.
Rights and permissions
About this article
Cite this article
Bourgain, J., Kahn, J., Kalai, G. et al. The influence of variables in product spaces. Israel J. Math. 77, 55–64 (1992). https://doi.org/10.1007/BF02808010
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02808010