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 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.

This is a corrigendum to the paper ‘A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems’ [3]. Theorem 7.1 in that paper is incorrect as stated, and the error originates with Proposition 7.5, part (iii), which was incorrectly quoted from a paper by Bergelson, Host, and Kra [1]. Consequently, this invalidates the proof of Theorem 4.2, which was used in the proofs of the main results in [3]. In this corrigendum we fix the problem by establishing a slightly weaker version of Theorem 7.1 (see §2 below) and use it to give a new proof of Theorem 4.2 (see §3 below). This ensures that all main results in [3] remain correct. We thank Zhengxing Lian and Jiahao Qiu for bringing this mistake to our attention.

Given a subshift $\Sigma $ of finite type and a finite set S of finite words, let $\Sigma \langle S\rangle $ denote the subshift of $\Sigma $ that avoids S. We establish a general criterion under which we can bound the entropy perturbation $h(\Sigma ) - h(\Sigma \langle S\rangle )$ from above. As an application, we prove that this entropy difference tends to zero with a sequence of such sets $S_1, S_2,\ldots $ under various assumptions on the $S_i$.