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.

中文翻译：

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Bergelson-Host-Kra 分解的谱细化和新的多重遍历定理 – 勘误表

这是对“Bergelson-Host-Kra 分解的谱细化和新的多重遍历定理”论文的更正[3]。正如所述，该论文中的定理 7.1 是不正确的，该错误源自命题 7.5 第 (iii) 部分，该命题被错误地引用自 Bergelson、Host 和 Kra [1] 的一篇论文。因此，这使得定理 4.2 的证明无效，该定理 4.2 用于证明 [3] 中的主要结果。在本勘误中，我们通过建立定理 7.1 的稍弱版本（参见下面的§2）来解决这个问题，并用它给出定理 4.2 的新证明（参见下面的§3）。这确保了[3]中的所有主要结果保持正确。我们感谢连正兴和邱家豪让我们注意到这个错误。