Journal d’Analyse Mathématique

, Volume 86, Issue 1, pp 183–220

An odd Furstenberg-Szemerédi theorem and quasi-affine systems


DOI: 10.1007/BF02786648

Cite this article as:
Host, B. & Kra, B. J. Anal. Math. (2002) 86: 183. doi:10.1007/BF02786648


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, andEX with μ(E) > 0, we show that there existsn ≡ j (modk) with ώ(ET-nE ∩T-2nE ∩T-3nE) > 0, so long asTk 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’.

Copyright information

© Hebrew University of Jerusalem 2002

Authors and Affiliations

  1. 1.Equipe d’analyse et de mathématiques appliquéesUniverité de Marne la ValléeMarne la Vallée CedexFrance
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA