Grid method for divergence of averages

Abstract

AbstractIn this paper, we will introduce the ‘grid method’ to prove that the extreme case of oscillation occurs for the averages obtained by sampling a flow along the sequence of times of the form $\{n^\alpha : n\in {\mathbb {N}}\}$ , where $\alpha $ is a positive non-integer rational number. Such behavior of a sequence is known as the strong sweeping-out property. By using the same method, we will give an example of a general class of sequences which satisfy the strong sweeping-out property. This class of sequences may be useful to solve a long-standing open problem: for a given irrational $\alpha $ , whether the sequence $(n^\alpha )$ is bad for pointwise ergodic theorem in $L^2$ or not. In the process of proving this result, we will also prove a continuous version of the Conze principle.