## Abstract

The influence of the kth coordinate on a Boolean function f : {0, 1}^{n} → {0, 1} is the probability (with respect to the uniform measure on {0, 1}^{n}) that flipping x_{k} changes the value f(x). The total influence I(f) is the sum of influences of the coordinates. The well-known ‘Junta theorem’ of Friedgut [Combinatorica 18 (1998) 27–35] asserts that if I(f) ≤ M,, then f can be ε-approximated by a function that depends on 2O(M/ε) coordinates. Friedgut's theorem has a wide variety of applications in mathematics and theoretical computer science. For a function Pr[f(x) = 1] = t ≤ 1/2,, the edge isoperimetric inequality on the cube implies that I(f) ≥ 2t log_{2}(1/t). Kahn and Kalai (2006) asked, in the spirit of the Junta theorem, whether any f such that I(f) is within a constant factor of the minimum, can be (εt) -approximated by a disjunctive normal form (DNF) of a ‘small’ size (that is, a union of a small number of subcubes). We answer the question by proving the following structure theorem: If I(f) ≤ 2t(log_{2}(1/t) +M),, then f can be (εt) -approximated by a DNF of size 2^{2O(M/ε)}. The dependence on M is sharp up to the constant factor in the double exponent.

Original language | American English |
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Pages (from-to) | 667-679 |

Number of pages | 13 |

Journal | Bulletin of the London Mathematical Society |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2018 |

Externally published | Yes |

### Bibliographical note

Funding Information:Received 16 July 2017; revised 7 March 2018; accepted 20 April 2018; published online 24 May 2018. 2010 Mathematics Subject Classification 05D40 (primary), 06E30, 05C35 (secondary). This research was supported by the Israel Science Foundation (grant nos. 402/13 and 1612/17) and by the Binational US-Israel Science Foundation (grant no. 2014290).

Publisher Copyright:

© 2018 London Mathematical Society

## Keywords

- 05C35 (secondary)
- 05D40 (primary)
- 06E30