An odd Furstenberg-Szemerédi theorem and Quasi-Affine Systems

Bernard Host*, Bryna Kra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We prove a version of Furstenberg's ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X, B, μ, T), integers 0 ≤ j < k, and E ⊂ X with μ(E) > 0, we show that there exists n ≡ j (mod k) with μ(E ∩ T-n E ∩ T-2 E ∩ T-3n E) > 0, so long as Tk is ergodic. This result requires a deeper understanding of the limit of some nonconventional ergodic averages and the introduction of a new class of systems, the 'Quasi-Affine Systems'.

Original languageEnglish (US)
Pages (from-to)183-220
Number of pages38
JournalJournal d'Analyse Mathematique
StatePublished - 2002

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)


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