A point of view on Gowers uniformity norms

Bernard Host*, Bryna Kra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of focusing on the norms themselves, we study associated dual norms and dual functions. Combining this study with a variant of the Szemerédi Regularity Lemma, we give a decomposition theorem for dual functions, linking the dual norms to classical norms and indicating that the dual norm is easier to understand than the norm itself. Using the dual functions, we introduce higher order algebras that are analogs of the classical Fourier algebra, which in turn can be used to further characterize the dual functions.

Original languageEnglish (US)
Pages (from-to)213-248
Number of pages36
JournalNew York Journal of Mathematics
StatePublished - Mar 28 2012


  • Dual functions
  • Dual norms
  • Gowers norms

ASJC Scopus subject areas

  • Mathematics(all)


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