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

TL;DR

This paper extends ergodic and combinatorial results involving measure preserving actions of affine semigroups to non-amenable cases using ultrafilter limits, revealing new {x+y, xy} pattern existence in number fields.

Contribution

It introduces a novel ultrafilter limit approach to analyze measure preserving actions of non-amenable affine semigroups, broadening previous results to more general algebraic structures.

Findings

Existence of {x+y, xy} patterns in any finite partition of number fields.

Extension of ergodic theorems to non-amenable affine semigroups.

Refinement of previous results using ultrafilter methods.

Abstract

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