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

Ergodic and combinatorial results obtained in [10] involved measure preserving actions of the affine group ${\mathcal{A}}_K$ 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 [10] to a more general situation involving the measure preserving actions of the non-amenable affine semigroups of a large class of integral domains. (The results in [10] 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 $i\in \{1,\dots ,r\}$ and many $x\in K$ and $y\in {\mathcal{O}}_K$ such that $\{x+y,xy\}\subset C_i$.