Behavior of ergodic averages along a subsequence and the grid method.

RUNAWAY SEMINAR 

Abstract: What is an ergodic theorem? What are space and time averages? Starting with this we will talk about a recent result on subsequence ergodic theorems we will talk about a recent result on subsequence ergodic theorem. It says that if $\alpha$ is a non-integer rational number, then for the `most functions' the time averages along $(n^\alpha)$ fail to converge almost everywhere. Moreover, an extreme case of non-convergence occurs in this case which is known as the strong sweeping out property.  This result is an improvement of a result of V. Bergelson, M. Boshernitzan and J. Bourgain which says that the averages along $(n^\alpha)$ diverge a.e. Note that the averages do converge in $L^2$ norm. The method we use is quite general and can be used to settle other open problems or improve other results.  In the talk we will mention some quite old  open problems.