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Anthony Ostuni

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3

FOCS Conference 2025 Conference Paper

Quasipolynomial Bounds for the Corners Theorem

  • Michael Jaber
  • Yang P. Liu
  • Shachar Lovett
  • Anthony Ostuni
  • Mehtaab Sawhney

Let G be a finite abelian group and A be a subset of $G \times G$ which is corner-free, meaning that there are no $x, y \in G$ and $d \in G \backslash\{0\}$ such that $(x, y), (x+d, y), (x, y+d) \in A$. We prove that \begin{equation*}|A| \leq|G|^{2} \cdot \exp \left(-(\log |G|)^{\Omega{1}}\right)\end{equation*}As a consequence, we obtain polynomial (in the input length) lower bounds on the non-deterministic communication complexity of Exactly-N in the 3-player Number-on-Forehead model. We also obtain the first “reasonable” lower bounds on the coloring version of the 3 -dimensional corners problem, as well as on the non-deterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model. This is an extended abstract. The full version of the paper can be found at https: //arxiv. org/abs/2504. 07006.

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