Undecidability of Polynomial Inequalities in Subset Densities and Additive Energies

TL;DR

This paper demonstrates that determining the validity of polynomial inequalities involving subset densities and additive energies in additive combinatorics is undecidable, extending known results from graph theory to this domain.

Contribution

It proves two theorems of undecidability for polynomial inequalities in additive combinatorics, showing their inherent computational difficulty.

Findings

Undecidability of polynomial inequalities in subset densities.

Extension of graph homomorphism density results to additive combinatorics.

Use of algebraic, graph construction, and Fourier analysis techniques.

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.