Revisiting sums and products in countable and finite fields

We establish a polynomial ergodic theorem for actions of the affine group of a countable field K . As an application, we deduce—via a variant of Furstenberg’s correspondence principle—that for fields of characteristic zero, any ‘large’ set E⊂ K contains ‘many’ patterns of the form (u)+v,uv\ , for every non-constant polynomial p(x)∈ K[x] . Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a finitistic variant of Bergelson’s ‘colouring trick’, show that for r∈ N fixed, any r -colouring of a large enough finite field will contain monochromatic patterns of the form ,p(u)+v,uv\ . In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ ,v,u+v,uv\ theorem.