默比乌斯函数与序列的正交关系

动力系统是一门研究系统长期演化行为的学科。它在分析、几何、拓扑、数论等众多数学核心领域中都有着重要的应用。沃尔夫数学奖得主Peter Sarnak于2010年提出了Möbius不相交性猜想。该猜想是完全从动力系统的角度去研究Möbius函数取值的随机性问题，引起了动力系统和数论等相关领域学者的广泛关注。本项目首先研究Sarnak猜想的“逆问题”，探讨正拓扑熵系统所对应的序列与Möbius函数之间的正交关系。其次，发展Mauduit、Rivat等人的工作，研究Bernoulli移位系统中的某些序列及其子序列，构造可与Möbius函数正交的正熵序列的新例子。最后，发展Vinogradov双线性方法、Matomäki–Radziwiłł–Tao平均值估计方法以及Tao的熵递减论证方法，研究具有不规则轨道分布或混合性更强的零拓扑熵系统所对应的确定性序列与Möbius函数之间的正交关系。

Dynamic systems theory deals with the long-term qualitative behavior of systems. It has important applications in many core areas of mathematics such as analysis, geometry, topology and number theory. Wolf Prize in Mathematics winner Peter Sarnak proposed the Möbius disjointness conjecture in 2010. This conjecture is focused on the randomness of the Möbius function under the perspective of dynamical systems, which has attracted much attention among the experts in the areas of dynamical systems and number theory. In this project, we will be interested in the “inverse” of Sarnak’s problem first. In particular, the orthogonal relation between the Möbius function and the sequences from the dynamical system with positive topological entropy will be explored. Furthermore, we will develop the estimate method due to Mauduit and Rivat, and study some sequences and subsequences from the Bernoulli shift. Moreover, the new examples of the sequences with positive entropy, which are Möbius orthogonal, will be obtained. Finally, we will generalize the Vinogradov’s bilinear method, the estimate by Matomäki–Radziwiłł–Tao and the entropy decrement argument due to Tao. Research on this project enables us to comprehend the orthogonal relation between the Möbius function and the sequences from general deterministic flows with irregular distribution of orbits and statistically more complexity.