## Abstract

A function fg¶{0,1}^{n}→ {0,1} is called an approximate AND-homomorphism if choosing x,ygn uniformly at random, we have that f(xg§ y) = f(x)g§ f(y) with probability at least 1-ϵ, where xg§ y = (x_{1}g§ y_{1},...,x_{n}g§ y_{n}). We prove that if fg¶ {0,1}^{n} → {0,1} is an approximate AND-homomorphism, then f is -close to either a constant function or an AND function, where (ϵ) → 0 as ϵ→ 0. This improves on a result of Nehama, who proved a similar statement in which δdepends on n. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if f is ϵ-close to satisfying judgement aggregation, then it is (ϵ)-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which δdecays polynomially with n. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation f = λ g, where is the downwards noise operator f(x) = _{y}[f(x g§ y)], f is [0,1]-valued, and g is {0,1}-valued. We identify all exact solutions to this equation, and show that any approximate solution in which f and λ g are close is close to an exact solution.

Original language | English |
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Title of host publication | STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy |

Publisher | Association for Computing Machinery |

Pages | 222-233 |

Number of pages | 12 |

ISBN (Electronic) | 9781450369794 |

DOIs | |

State | Published - 8 Jun 2020 |

Externally published | Yes |

Event | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States Duration: 22 Jun 2020 → 26 Jun 2020 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 |
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Country/Territory | United States |

City | Chicago |

Period | 22/06/20 → 26/06/20 |

### Bibliographical note

Publisher Copyright:© 2020 ACM.

### Funding

∗This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC. †Supported partially by NSF grant CCF-1412958 and Rothschild Fellowship. ‡Supported by NSF Award DMS-1737944, by AROMURI W911NF1910217 and by a Simons Investigator Award in Mathematics (622132).

Funders | Funder number |
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AROMURI | W911NF1910217, 622132 |

National Science Foundation | DMS-1737944, CCF-1412958, 1737944, 1412958, 802020 |

Horizon 2020 Framework Programme |

## Keywords

- Analysis of Boolean Functions
- Property Testing