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

Published online by Cambridge University Press:  26 July 2016

VITALY BERGELSON  and
JOEL MOREIRA
VITALY BERGELSON
Affiliation:
Department of Mathematics, The Ohio State University, Math Building, 231 West 18th Avenue, Columbus, OH 43201, USA email vitaly@math.ohio-state.edu, moreira@math.ohio-state.edu
JOEL MOREIRA
Affiliation:
Department of Mathematics, The Ohio State University, Math Building, 231 West 18th Avenue, Columbus, OH 43201, USA email vitaly@math.ohio-state.edu, moreira@math.ohio-state.edu
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Abstract

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 the non-amenable affine 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 $y\in {\mathcal{O}}_{K}$, $\{x+y,xy\}\subset C_{i}$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 473 - 498
Copyright
© Cambridge University Press, 2016 

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