Undecidability of Polynomial Inequalities in Subset Densities and Additive Energies

2025-05-12

AI Summary

This paper addresses the Turing decidability of polynomial inequalities involving subset densities and additive energy in additive combinatorics: given an inequality of the form $p(delta_A, delta_B, dots) geq 0$, does it hold universally for all subsets of all finite abelian groups? Extending the Hatami–Norine method for undecidability of graph density inequalities to additive combinatorics, the authors employ a deep synthesis of algebraic constructions, graph embeddings, and discrete Fourier analysis to prove that this decision problem is Turing-undecidable in general. Their main contributions are threefold: (1) establishing the undecidability of polynomial inequalities in subset densities; (2) concurrently proving the undecidability of analogous inequalities involving additive energy; and (3) uncovering the intrinsic computational complexity of density problems in additive combinatorics, thereby forging profound connections among subset density, Fourier-analytic methods, and algebraic structure.

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

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Many results in extremal graph theory can be formulated as certain polynomial inequalities in graph homomorphism densities. Answering fundamental questions raised by Lov{'a}sz, Szegedy and Razborov, Hatami and Norine proved that determining the validity of an arbitrary such polynomial inequality in graph homomorphism densities is undecidable. We observe that many results in additive combinatorics can also be formulated as polynomial inequalities in subset's density and its variants. Based on techniques introduced in Hatami and Norine, together with algebraic and graph construction and Fourier analysis, we prove similarly two theorems of undecidability, thus showing that establishing such polynomial inequalities in additive combinatorics are inherently difficult in their full generality.

Problem

• Proves undecidability of polynomial inequalities in subset densities
• Extends undecidability results to additive combinatorics problems
• Demonstrates inherent difficulty in general polynomial inequality validation

Innovation

• Techniques from Hatami and Norine
• Algebraic and graph construction
• Fourier analysis methods