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arXiv · 2311.14198

Saturated theorem along cubes for a measure and applications

Jiahao Qiu·Jiaqi Yu
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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 $k'\leqslant k$. 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].

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Jiahao Qiu, Jiaqi Yu. 2023-11-23. Saturated theorem along cubes for a measure and applications. https://arxiv.org/abs/2311.14198

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