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Uniformity norms, their weaker versions, and applications

Volume 203 / 2022

Pandelis Dodos, Vassilis Kanellopoulos Acta Arithmetica 203 (2022), 251-270 MSC: Primary 11B30. DOI: 10.4064/aa210728-27-2 Published online: 10 May 2022


We show that, under some mild hypotheses, the Gowers uniformity norms (both in the additive and in the hypergraph setting) are essentially equivalent to certain weaker norms which are easier to understand. We present two applications of this equivalence: a variant of the Koopman–von Neumann decomposition, and a proof of the relative inverse theorem for the Gowers $U^s[N]$-norm using a norm-type pseudorandomness condition.


  • Pandelis DodosDepartment of Mathematics
    University of Athens
    157 84, Athens, Greece
  • Vassilis KanellopoulosDepartment of Mathematics
    Faculty of Applied Sciences
    National Technical University of Athens
    Zografou Campus
    157 80, Athens, Greece

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