仿射半群的保测度作用与{x+y，xy}模式
Measure preserving actions of affine semigroups and {x+y,xy} patterns
摘要：
在[10]中得到的遍历和组合结果涉及可数域$K$的仿射群${\mathcal A}_K$的保测度作用。本文发展了一种基于超滤子极限的新方法，它使文[10]中的结果得到改进和推广，使之适用于一类整环的非顺从仿射半群的测度保持作用的更一般的情形。(The[10]中的结果很大程度上依赖于域的仿射群的顺从性）。除其他事项外，我们得到，作为超滤遍历定理的推论，以下结果：令PDMATHAMN是数域，让${\mathcal O}_K$是整数环$K$。对任意有限划分$K=C_1\cup\cdots\cup C_r$，存在$i\in\{1,\dots,r\}$和多个$x\in K$和$y\in{\mathcal O}_K$，使得$\{x+y,xy\}\subset C_i$ .
英文摘要：
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 $\{x+y,xy\}\subset C_i$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
