Mixing and double recurrence in probability groups

We define a class of groups equipped with an invariant probability measure, which includes all compact groups and is closed under taking ultraproducts with the induced Loeb measure. We call these probability groups and develop the basics of the theory of their measure-preserving actions on probability spaces, including a natural notion of mixing. A short proof reveals that for probability groups mixing implies double recurrence, which generalizes a theorem of Bergelson and Tao proved for ultraproducts of finite groups. Moreover, a quantitative version of our proof gives that $\epsilon$-approximate mixing implies $3\sqrt{\epsilon}$-approximate double recurrence. Examples of approximately mixing probability groups are quasirandom groups introduced by Gowers, so the last theorem generalizes and sharpens the corresponding results for quasirandom groups of Bergelson and Tao, as well as of Austin.Comment: Corrected the definitions of probability groups and their actions. Added a quick overview of ultraproducts and their countable-compactness, as well as the Loeb measure constructio

Pointwise ergodic theorem for locally countable quasi-pmp graphs

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, analogous to pointwise ergodic theorems for group actions, replacing the group with a Schreier graph of the action. For any quasi-pmp graph, the theorem gives an increasing sequence of Borel subgraphs with finite connected components along which the averages of $L^1$ functions converge to their expectations. Equivalently, it states that any (not necessarily pmp) locally countable Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is descriptive set theoretic and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant, a method of producing finite equivalence subrelations with large domain, and a simple method of exploiting nonamenability of a measured graph. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.Comment: Added to the introduction a discussion of existing results about pointwise ergodic theorems for quasi-action

Upper bounds for alpha-domination parameters

In this paper, we provide a new upper bound for the alpha-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the alpha-rate domination number, which combines the concepts of alpha-domination and k-tuple domination.Comment: 7 pages; Presented at the 4th East Coast Combinatorial Conference, Antigonish (Nova Scotia, Canada), May 1-2, 200