A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems – CORRIGENDUM

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.