A saturated theorem along cubes for a measure and applications

Abstract We show that for a minimal system $(X,T)$ , the set of saturated points along cubes with respect to its maximal $\infty $ -step pro-nilfactor $X_\infty $ has a full measure. As an application, it is shown that if a minimal system $(X,T)$ has no non-trivial $(k+1)$ -tuples with arbitrarily long finite IP-independence sets, then it has only at most k ergodic measures and is an almost $k'$ to one extension of $X_\infty $ for some $k'\leqslant k$ . In particular, for $k=1$ , we prove that $(X,T)$ is uniquely ergodic (even regular with respect to $X_\infty $ ), which answers a conjecture stated by Dong et al [Infinite-step nilsystems, independence and complexity. Ergod. Th. & Dynam. Sys. 33 (1) (2013), 118–143].