  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
$\{x+y,xy\}\subset C_i$.
