## Abstract

Given as input two n-element sets A, B ⊆ {0, 1}d with d = c log n ≤ (log n)2/(log log n)4 and a target t ∈ {0, 1,⋯, d}, we show how to count the number of pairs (x, y) ∈ A × B with integer inner product 〉x, y〈 = t deterministically, in n2/2Ω(√log n log log n/(c log2 c)) time. This demonstrates that one can solve this problem in deterministic subquadratic time almost up to log2 n dimensions, nearly matching the dimension bound of a subquadratic randomized detection algorithm of Alman and Williams [FOCS 2015]. We also show how to modify their randomized algorithm to count the pairs w.h.p., to obtain a fast randomized algorithm. Our deterministic algorithm builds on a novel technique of reconstructing a function from sum-aggregates by prime residues, or modular tomography, which can be seen as an additive analog of the Chinese Remainder Theorem. As our second contribution, we relate the fine-grained complexity of the task of counting of vector pairs by inner product to the task of computing a zero-one matrix permanent over the integers.

Original language | English |
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Title of host publication | 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021 |

Editors | Nikhil Bansal, Emanuela Merelli, James Worrell |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Number of pages | 21 |

ISBN (Electronic) | 9783959771955 |

DOIs | |

Publication status | Published - 1 Jul 2021 |

MoE publication type | A4 Article in a conference publication |

Event | International Colloquium on Automata, Languages, and Programming - Virtual, Online, Glasgow, United Kingdom Duration: 12 Jul 2021 → 16 Jul 2021 Conference number: 48 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Publisher | Dagstuhl Publishing |

Volume | 198 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | International Colloquium on Automata, Languages, and Programming |
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Abbreviated title | ICALP |

Country/Territory | United Kingdom |

City | Glasgow |

Period | 12/07/2021 → 16/07/2021 |

## Keywords

- Additive reconstruction
- Chinese Remainder Theorem
- Counting
- Inner product
- Modular tomography
- Orthogonal vectors
- Permanent