## Abstract

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is o(log n). However, both results become useless when the total influence of the function is ω(log n). The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of S_{n}. In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: 1.A quantitative improvement of the Bourgain–Kalai result regarding the total influence of functions that are transitively symmetric.2.A slightly weaker version of the Fourier-Entropy Conjecture of Friedgut and Kalai. Our result establishes new bounds on the Fourier entropy of a Boolean function f, as well as stronger bounds on the Fourier entropy of low-degree parts of f. In particular, it implies that the Fourier spectrum of a constant variance, Boolean function f is concentrated on 2 ^{O}^{(}^{I}^{[}^{f}^{]}^{log}^{I}^{[}^{f}^{]}^{)} characters, improving an earlier result of Friedgut. Removing the log I[f] factor would essentially resolve the Fourier-Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas. Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time 2 ^{O}^{(}^{K}^{log}^{K}^{)} using membership queries. Thus, the class of functions with total influence O(log n/ log log n) is agnostically learnable in poly(n) time.

Original language | English |
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Pages (from-to) | 1097-1138 |

Number of pages | 42 |

Journal | Geometric and Functional Analysis |

Volume | 30 |

Issue number | 4 |

State | Published - 1 Aug 2020 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2020, Springer Nature Switzerland AG.

### Funding

Dor Minzer: Supported partially by NSF Grant DMS-1638352 and Rothschild Fellowship. Muli Safra: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 835152).

Funders | Funder number |
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National Science Foundation | DMS-1638352 |

Horizon 2020 Framework Programme | 835152 |

European Research Council |