Measure preserving actions of affine semigroups and {x+y,xy} patterns

Ergodic and combinatorial results obtained in Bergelson and Moreira [Ergodic theorem involving additive and multiplicative groups of a field and {x + y, x y} patterns. Ergod. Th. & Dynam. Sys. to appear, published online 6 October 2015, doi:10.1017/etds. 2015.68], involved measure preserving actions of the affine group of a countable field K. In this paper, we develop a new approach, based on ultrafilter limits, which allows one to refine and extend the results obtained in Bergelson and Moreira, op. cit., to a more general situation involving measure preserving actions of the non-amenable affine semigroups of a large class of integral domains. (The results and methods in Bergelson and Moreira, op. cit., heavily depend on the amenability of the affine group of a field.) Among other things, we obtain, as a corollary of an ultrafilter ergodic theorem, the following result. Let K be a number field and let OK be the ring of integers of K. For any finite partition K = C₁ ⊂ ... ⊂ C_r , there exists i ϵ {1, ... , r} such that, for many x ϵ K and many y ϵ O_K , {x + y, x y} ⊂ C_i .