Cut and project schemes, combinatorics and averaging properties for Toeplitz subshifts

该项目涉及与Toeplitz词相关的子移位的各种性质。这里的“词”是从整数到有限集合的映射，也就是符号的无限连接。在Toeplitz的话中，这种连接是以这样一种方式构建的，即单词不是周期性的，但仍然表现出一定程度的顺序。这使得它们成为非周期有序理论和有趣的准晶一维模型的重要例子。在准晶的背景下，相关薛定谔算符的光谱扮演着重要的角色，因为它编码了电子可用的能级。如果满足所谓的前导序列条件(LSC)，则谱是勒贝格测度零的康托集(实际上它甚至意味着局部恒定余周期的一致性)。然而，到目前为止，LSC只显示了所谓的简单Toeplitz词的子移位。该项目的一个目标是将这一证明扩展到更广泛的Toeplitz词类别。此外，还计划找到一般Toeplitz词满足LSC的充分或必要条件。LSC的部分内容与所涉及的词的组合性质有关。在这里，重要的一步是研究与Toeplitz词有关的切割和投影方案。它们提供了一种从周期的高维晶格生成非周期的一维单词的方法。因此，了解单词的组合属性如何与投影集的属性相关是项目的第一个目标；分析组合属性本身是第二个目标。LSC的剩余部分处理沿Toeplitz单词的矩阵值函数的平均，但也计划调查实值函数。这个话题特别有趣，因为最近的一个预印本将被广泛研究的Sarnak猜想简化为对Toeplitz词的“高阶”(熵为零)的平均。这个猜想涉及到所谓的莫比乌斯函数的平均值，它在数论中扮演着重要的角色。

The project concerns various properties of subshifts associated to Toeplitz words. Here a "word" is a map from the integers into a finite set, that is, an infinite concatenation of symbols. In Toeplitz words, this concatenation is constructed in such a way that the word is not periodic, but still exhibits a certain degree of order. This makes them important examples in the theory of aperiodic order and interesting one-dimensional models for quasicrystals.In the context of quasicrystals, the spectrum of the associated Schrödinger operator plays an important role, since it encodes the energy levels that are available to electrons. If the so-called leading sequence condition (LSC) is satisfied, then the spectrum is a Cantor set of Lebesgue measure zero (actually it even implies uniformity of locally constant cocycles). However, the LSC has so far only been shown for subshifts of so-called simple Toeplitz words. One aim of the project is to extend this proof to a wider class of Toeplitz words. In addition it is planned to find sufficient or necessary conditions for general Toeplitz words to satisfy the LSC.Parts of the LSC are related to combinatorial properties of the involved words. Here an important step is to study cut and project schemes which can be associated to the Toeplitz word. They provide a way to generate the non-periodic, one-dimensional words from periodic, higher-dimensional lattices. Understanding how combinatorial properties of the word are related to properties of the projection set is therefore a first goal of the project; analysing the combinatorial properties themselves is a second one.The remaining part of the LSC deals with averages of matrix-valued functions along Toeplitz words, but the investigation of real-valued functions is planned as well. This topic is particularly interesting, since a recent preprint reduced the much studied Sarnak conjecture to averages over Toeplitz words with "high order" (entropy zero). The conjecture concerns averages of the so-called Möbius function, which play an important role in number theory.