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On Generalized Corners and Matrix Multiplication

Author Kevin Pratt

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Kevin Pratt
  • Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, NY, USA

Acknowledgements

I thank Ryan O'Donnell for many useful discussions about these problems. I also thank Chris Umans for making comments which motivated this paper early on, and in particular, which motivated Theorem 30. I thank the reviewers for suggestions which improved the exposition of this paper.

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Kevin Pratt. On Generalized Corners and Matrix Multiplication. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 89:1-89:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.89

Abstract

Suppose that S ⊆ [n]² contains no three points of the form (x,y), (x,y+δ), (x+δ,y'), where δ ≠ 0. How big can S be? Trivially, n ≤ |S| ≤ n². Slight improvements on these bounds are obtained from Shkredov’s upper bound for the corners problem [Shkredov, 2006], which shows that |S| ≤ O(n²/(log log n)^c) for some small c > 0, and a construction due to Petrov [Fedor Petrov, 2023], which shows that |S| ≥ Ω(n log n/√{log log n}).
Could it be that for all ε > 0, |S| ≤ O(n^{1+ε})? We show that if so, this would rule out obtaining ω = 2 using a large family of abelian groups in the group-theoretic framework of [Cohn and Umans, 2003; Cohn et al., 2005] (which is known to capture the best bounds on ω to date), for which no barriers are currently known. Furthermore, an upper bound of O(n^{4/3 - ε}) for any fixed ε > 0 would rule out a conjectured approach to obtain ω = 2 of [Cohn et al., 2005]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Algebraic computation
  • fast matrix multiplication
  • additive combinatorics

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