Quasipolynomial bounds for the corners theorem

2025-04-09

AI Summary

This paper investigates the maximum density of “corner-free sets” in the two-dimensional grid over a finite abelian group $ G $—i.e., subsets $ A subseteq G imes G $ containing no triple of the form $ (x,y), (x+d,y), (x,y+d) $ for any $ d eq 0 $. Employing a synthesis of high-dimensional Fourier analysis, additive combinatorics, and extremal methods, the authors establish the first effective lower bound for the coloring variant of the three-dimensional corner problem. This yields the upper bound $ |A| leq |G|^2 cdot expig(-(log |G|)^{Omega(1)}ig) $, demonstrating that corner-free density decays quasi-polynomially. The result leads to two breakthroughs in communication complexity: (1) the first polynomial lower bound on the *nondeterministic* communication complexity of the Exactly-$N$ function in the 3-party Number-on-Forehead (NOF) model; and (2) the first nontrivial lower bound on its *deterministic* communication complexity in the 4-party NOF model.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $G$ be a finite abelian group and $A$ be a subset of $G imes G$ which is corner-free, meaning that there are no $x, y in G$ and $d in G setminus {0}$ such that $(x, y)$, $(x+d, y)$, $(x, y+d) in A$. We prove that [|A| le |G|^2 cdot exp(-(log |G|)^{Omega(1)}).] 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 and equivalently the deterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model.

Problem

• Bounds corner-free subsets in abelian groups
• Lower bounds nondeterministic communication complexity Exactly-N
• Lower bounds deterministic communication complexity Exactly-N

Innovation

• Corner-free subset bounds in abelian groups
• Non-deterministic communication complexity lower bounds
• Coloring version of 3D corners problem