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Failure of the Roth theorem for solvable groups of exponential growth

Published online by Cambridge University Press:  02 February 2004

V. BERGELSON  and
A. LEIBMAN
V. BERGELSON
Affiliation:
Department of Mathematics, The Ohio State University, 100, Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (e-mail: vitaly@math.ohio-state.edu, leibman@math.ohio-state.edu)
A. LEIBMAN
Affiliation:
Department of Mathematics, The Ohio State University, 100, Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (e-mail: vitaly@math.ohio-state.edu, leibman@math.ohio-state.edu)
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Abstract

We show that for any finitely generated solvable group of exponential growth one can find a measure-preserving action for which the multiple recurrence theorem fails and a measure-preserving action for which the ergodic Roth theorem fails. This is in contrast with the positive results established by Leibman (Geom. Funct. Anal.8 (1998), 853–931) and Bergelson and Leibman (Invent. Math.147 (2002), 429–470) for nilpotent group actions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 1 , February 2004 , pp. 45 - 53
Copyright
2004 Cambridge University Press

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