Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann’s mean and Birkhoff’s pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of being eventually totally bounded in the mean is necessary and sufficient. This condition involves ...

Shortly after Szemerédi’s proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive combinatorics are addressed kwith ergodic theory. Combinatorial ergodic theory has since produced combinatori...