Ergodic and combinatorial results obtained in Bergelson and Moreira [Ergodic theorem involving additive and multiplicative groups of a field and$\{x+y,xy\}$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 thenon-amenableaffine 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${\mathcal{O}}_{K}$be the ring of integers of$K$. For any finite partition$K=C_{1}\cup \cdots \cup C_{r}$, there exists$i\in \{1,\ldots ,r\}$such that, for many$x\in K$and many$y\in {\mathcal{O}}_{K}$,$\{x+y,xy\}\subset C_{i}$.
Unital affine semigroups