A saturated theorem along cubes for a measure and applications

We show that for a minimal system $(X,T)$ , the set of saturated points along cubes with respect to its maximal ∞ -step pro-nilfactor X_∞ 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_∞ for some k' k . In particular, for $k=1$ , we prove that $(X,T)$ is uniquely ergodic (even regular with respect to X_∞ ), which answers a conjecture stated by Dong et al [Infinite-step nilsystems, independence and complexity. Ergod. Th. & Dynam. Sys. 33 (1) (2013), 118–143].