## Abstract

A Boolean function f: {0,1}^{n} → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure μ_{p} on the discrete cube, as long as log1/p≪logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n ^{-c} for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand's lemma, which easily generalizes in various directions, including non-monotone functions.

Original language | English |
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Pages (from-to) | 45-71 |

Number of pages | 27 |

Journal | Combinatorica |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2013 |

Externally published | Yes |

### Bibliographical note

Funding Information:∗ Partially supported by the Adams Fellowship Program of the Israeli Academy of Sciences and Humanities and by the Koshland Center for Basic Research. † Supported by the Israel Science Foundation and by the Binational Science Foundation.