Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits Jane Tunstill These are all critical thinking

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Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits Jane Tunstill These are all critical thinkingWe study a class of measure theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host Kra type seminorms for measure preserving actions of countable nilpotent groups. This yields an extension of