An uncountable ergodic Roth theorem and applications

<p style='text-indent:20px;'>We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and space need separable. This generalizes a previous result of Bergelson, McCutcheon Zhang, complements Zorin-Kranich. We following two additional results: First, combinatorial application about triangular patterns certain subsets Cartesian square arbitrary groups, extending Zhang for groups. Second, new uniformity aspect double recurrence theorem <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-systems uniformly groups id="M2">\begin{document}$ $\end{document}</tex-math></inline-formula>. Our crucial proof both these results.</p>