Undecidability of Polynomial Inequalities in Subset Densities and Additive Energies

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.