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arXiv · 2404.07380

Strong Bounds for Skew-Corner-Free Sets

Michael Jaber·Shachar Lovett·Anthony Ostuni
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Abstract

Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove any skew corner-free set has size at most $\exp(-\Omega(\log^{1/12} n))\cdot n^2$, nearly matching the best known lower bound of $\exp(-O(\sqrt{\log n}))\cdot n^2$ by Beker (arXiv'24). Our techniques generalize those of Kelley and Meka's recent breakthrough on three-term arithmetic progression (FOCS'23), answering a question of Beker (arXiv'24). We note that a similar bound was obtained concurrently and independently by Mili\'cevi\'c (arXiv'24).

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Michael Jaber, Shachar Lovett, Anthony Ostuni. 2024-04-10. Strong Bounds for Skew-Corner-Free Sets. https://arxiv.org/abs/2404.07380

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