Multiple recurrence and infinite measure preserving odometers (Q1282266)

\textit{H. Furstenberg} [J. Anal. Math. 31, 204-256 (1977; Zbl 0347.28016)] showed that every finite measure-preserving transformation exhibits multiple recurrence, a profound generalization of Poincaré recurrence. In this note the case of infinite measure-preserving systems is shown to be entirely different. Using an odometer construction, for every \(p>1\) an ergodic infinite measure-preserving system is found that is \(p\)-fold recurrent but not \((p+1)\)-fold recurrent.