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

该项目关注与Toeplitz词相关的子移位的各种属性。这里的“字”是从整数到有限集合的映射，也就是说，符号的无限连接。在Toeplitz的单词中，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有序性。这使得它们成为非周期序理论和准晶有趣的一维模型中的重要例子。在准晶的背景下，相关薛定谔算符的光谱起着重要的作用，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），则该谱是勒贝格测度为零的Cantor集（实际上它甚至包含局部常数上圈的一致性）。然而，到目前为止，LSC只显示了所谓的简单Toeplitz字的子移位。该项目的一个目的是将这个证明扩展到更广泛的一类Toeplitz词。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词的组合性质有关。这里的一个重要步骤是研究可以与Toeplitz词相关联的切割和项目方案。它们提供了一种从周期性的、高维的格子中生成非周期性的、一维的字的方法。因此，理解单词的组合属性如何与投影集的属性相关是该项目的第一个目标;分析组合属性本身是第二个目标。LSC的其余部分处理沿着Toeplitz单词的矩阵值函数的平均值，但也计划对实值函数进行调查。这个主题特别有趣，因为最近的预印本减少了研究很多的Sarnak猜想的平均数超过Toeplitz词与“高阶”（熵零）。该猜想涉及所谓的莫比乌斯函数的平均值，它在数论中起着重要作用。

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.