<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 squa...

In this note we give a short proof of a pointwise ergodic theorem for measure preserving actions of word hyperbolic groups, also obtained recently by Bufetov, Khristoforov and Klimenko. Our approach also applies to infinite measure spaces and one application is to linear actions of discrete groups on the plane. 0. Introduction The well-known Birkhoff ergodic theorem for ergodic transformations ...