TY - JOUR
T1 - An odd Furstenberg-Szemerédi theorem and Quasi-Affine Systems
AU - Host, Bernard
AU - Kra, Bryna
PY - 2002
Y1 - 2002
N2 - 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'.
AB - 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'.
UR - https://www.scopus.com/pages/publications/0036352027
UR - https://www.scopus.com/inward/citedby.url?scp=0036352027&partnerID=8YFLogxK
U2 - 10.1007/BF02786648
DO - 10.1007/BF02786648
M3 - Article
AN - SCOPUS:0036352027
SN - 0021-7670
VL - 86
SP - 183
EP - 220
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
ER -