## Abstract

Let X be a family of k-element subsets of [n] and let {f_{s}:s→Σ: s∈ X} be an ensemble of local functions, each defined over a subset s⊂ [n]. Is there a global function G:[n]→Σ such that f_{s} = G|_{s} for all s∈ X ? An agreement test is a randomized property tester for this question. One such test is the V-test, that chooses a random pair of sets s_{1},s_{2}∈ X with prescribed intersection size and accepts if f_{s1},f_{s2} agree on the elements in s_{1}∩ s_{2}. The low acceptance (or 1%) regime is concerned with the situation that the test succeeds with low but non-negligible probability Agree({f_{s}}) ≥ ϵ>0. A "classical"low acceptance agreement theorem says Agree (Formula presented) Such statements are motivated by PCP questions. The case X= (_{k}^{[n]}) is well-studied and known as "direct product testing", which is related to the parallel repetition theorem. Finding sparser families X that satisfy (∗) is known as derandomized direct product testing. Prior to this work, the sparsest family satisfying (∗) had |X|≈ n^{25}, and we show X with |X|≈ n^{2}. We study the general behavior of high dimensional expanders with respect to agreement tests in the low acceptance regime. High dimensional expanders, even very sparse ones with |X|=O(n), are known to satisfy the high acceptance variant (where ϵ =1-o(1)). It has been an open challenge to analyze the low acceptance regime. Surprisingly, topological covers of X play an important role. We show that: If X has no connected covers, then (∗) holds, provided that X satisfies an additional expansion property, called swap cosystolic expansion. If X has a connected cover, then (∗) fails. If X has a connected cover (and swap-cosystolic-expansion), we replace (∗) by a statement that takes covers into account:(Formula presented). The property of swap-cosystolic-expansion holds for quotients of the Bruhat Tits buildings. As a corollary we derive (∗) for X being a spherical building, yielding a derandomized family with |X| ≈ n^{2}. We also derive (∗∗) for LSV complexes X, for which |X|=O(n).

Original language | English |
---|---|

Title of host publication | STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing |

Editors | Bojan Mohar, Igor Shinkar, Ryan O�Donnell |

Publisher | Association for Computing Machinery |

Pages | 1967-1977 |

Number of pages | 11 |

ISBN (Electronic) | 9798400703836 |

DOIs | |

State | Published - 10 Jun 2024 |

Externally published | Yes |

Event | 56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada Duration: 24 Jun 2024 → 28 Jun 2024 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
---|---|

ISSN (Print) | 0737-8017 |

### Conference

Conference | 56th Annual ACM Symposium on Theory of Computing, STOC 2024 |
---|---|

Country/Territory | Canada |

City | Vancouver |

Period | 24/06/24 → 28/06/24 |

## Keywords

- Agreement
- Agreement Testing
- Covers
- Direct Product Testing
- HDX
- High Dimensional Expanders

## ASJC Scopus subject areas

- Software