## Abstract

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. A central step in our analysis of quadraticity tests is the proof of aninverse theorem for the third Gowers uniformity norm of boolean functions. The last result implies that it ispossible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius. Our main technical tools are Fourier analysis on Z _{2} ^{n} and methods from additive number theory. We observe that these methods can be used to give a tight analysis of the Abelian Homomorphism testing problemfor some families of groups, including powers of Z _{p}.

Original language | American English |
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Title of host publication | STOC'07 |

Subtitle of host publication | Proceedings of the 39th Annual ACM Symposium on Theory of Computing |

Pages | 506-515 |

Number of pages | 10 |

DOIs | |

State | Published - 2007 |

Event | STOC'07: 39th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: 11 Jun 2007 → 13 Jun 2007 |

### Publication series

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

### Conference

Conference | STOC'07: 39th Annual ACM Symposium on Theory of Computing |
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Country/Territory | United States |

City | San Diego, CA |

Period | 11/06/07 → 13/06/07 |

### Bibliographical note

Funding Information:We would like to thank Jay Anderson for providing us with his suite of PSF-fitting and image alignment software, and for his valuable instruction, guidance and technical support. We also thank the anonymous referee for a constructive review. Support for this work was provided by NASA grants GO-13390 and GO-13791 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute. This work has made use of data from the European Space Agency (ESA) mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

## Keywords

- Low-degree tests