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

arXiv admin note: text overlap with arXiv:2202.08782; text overlap with arXiv:1105.3584, arXiv:1007.0189 by other authors