Multiple recurrence of Markov shifts and other infinite measure preserving transformations (Q1567348)

This paper considers an ergodic version of the following conjecture of Erdős: \[ K\subseteq \mathbb{Z}^+,\quad \sum_{n\in K}{1\over n}= \infty\;\Rightarrow\;K\text{ contains arithmetic progressions of all lengths}. \] The conjecture applies to infinite measure-preserving transformations, and it is shown that a stronger version of the conjecture is true for Markov shifts, but the stronger version is not true for all measure-preserving transformations. For a decreasing sequence \(\{c_n\}\), a conservative, ergodic measure-preserving transformation \(T\) is said to be \(\{c_n\}\)-\textit{conservative} if \(\sum^\infty_{n=1} c_nf\circ T^n= \infty\) a.e. for all \(f\in L^1_+\). The ergodic version of the Erdős question is that \(\{1/n\}\)-conservative, ergodic measure-preserving transformations are multiply recurrent. The Erdős conjecture implies the ergodic version, but it is unknown if the converse is true.