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On Generalized Corners and Matrix Multiplication

Author Kevin Pratt

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

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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.

Cite As Get BibTex

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

Author Details

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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