Grid method for divergence of averages

TL;DR

This paper introduces the grid method to analyze oscillation and divergence of averages along specific sequences, demonstrating the strong sweeping out property and its implications for sequences like $n^eta$.

Contribution

The paper develops the grid method to prove divergence properties of averages along sequences and introduces a continuous Conze principle, advancing understanding of oscillation phenomena.

Findings

Sequences of the form $n^eta$ exhibit strong sweeping out behavior.

The grid method effectively proves divergence of averages for certain sequences.

A continuous version of the Conze principle is established.

Abstract

In 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 the longstanding open problem: for a given irrational $\alpha$, whether the sequence $(n^\alpha)$ is `pointwise bad' for $L^p$ or not. In the process of proving these results, we will prove a continuous version of the Conze principle.