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 $\mathrm{\infty }$-step pro-nilfactor $X_{\mathrm{\infty }}$ 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 $k^{\mathrm{\prime }}$ to one extension of $X_{\mathrm{\infty }}$ for some $k^{\mathrm{\prime }}⩽k$. Particularly, for $k=1$ we prove that $(X,T)$ is uniquely ergodic (even regular with respect to $X_{\mathrm{\infty }}$), which answers a conjecture stated in [3].