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 \p (x) +y, xy\, for every non-constant polynomial p (x) Kx. 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 new 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 \x, p (x) +y, xy\. 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 generalisation of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic "colouring trick", we provide a conditional, elementary generalisation of Green and Sanders' \x, y, x+y, xy\ theorem.