Analysis of two step nilsequences

Host, Bernard; Kra, Bryna

Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1407-1453.

Les nilsuites sont apparues dans l’étude des moyennes ergodiques multiples associées à la démonstration par Furstenberg du théorème de Szemerédi. Depuis, elles ont aussi joué un rôle dans des questions de combinatoire additive. Les nilsuites sont une généralisation des suites presque périodiques et nous déterminons quelles parties de la théorie des suites presque périodiques peuvent s’étendre aux nilsuites d’ordre deux. Nous établissons les propriétés de base de ces suites et donnons une classification.

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg’s proof of Szemerédi’s Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for two step nilsequences and give a classification scheme for them.

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