Pointwise ergodic theorem along a subsequence of integers

After Birkhoff’s Pointwise Ergodic Theorem was proved in 1931, there have been many attempts to generalize the theorem along a subsequence of the integers instead of taking the entire se- quence (n). In this talk, we will present the following result of Roger Jones and Máté Wierdl:
If a sequence (an) satisfies an+1/an ≥ +1 + 1/(log n)12−ε , for some ε > 0, then in any aperiodic dynamical system (X, Σ, μ, T), we can always find a function f ∈ L2 such that the Cesàro averages along the se- quence (an) which is defined by An∈[N] f (Tan x) := N1 ∑ f (Tan x) (0.1) n∈[N] fail to converge in a set of positive measure.