A point of view on Gowers uniformity norms
Résumé
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 Szemer\'edi 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.