论克里斯托弗话语的叠加
On the superimposition of Christoffel words
摘要：
弗伦克尔猜想最初是用比蒂序列来表述的，可以重新表述如下：对于一个$k$ -字母表A，有一个固定的$k \geq 3$，存在一个唯一的平衡无限字，直到字母排列和移位，它有相互不同的字母频率。受Fraenkel猜想的启发，本文研究了两个Christoffel词是否可以叠加。在前人工作的基础上，利用Beatty序列，给出了两个等长的Christoffel字，更一般地，两个任意Christoffel字的叠加的充要条件.此外，对于任何两个可叠加的Christoffel词，我们给出了不同的可能叠加的数量，我们证明了存在一个叠加，适用于任何两个可叠加的Christoffel词。最后，得到了Christoffel词的一些新性质，并利用Christoffel词给出了一个关于货币问题的经典结果的几何证明.
英文摘要：
Initially stated in terms of Beatty sequences, the Fraenkel conjecture can be reformulated as follows: for a $k$-letter alphabet A, with a fixed $k \geq 3$, there exists a unique balanced infinite word, up to letter permutations and shifts, that has mutually distinct letter frequencies. Motivated by the Fraenkel conjecture, we study in this paper whether two Christoffel words can be superimposed. Following from previous works on this conjecture using Beatty sequences, we give a necessary and sufficient condition for the superimposition of two Christoffel words having same length, and more generally, of two arbitrary Christoffel words. Moreover, for any two superimposable Christoffel words, we give the number of different possible superimpositions and we prove that there exists a superimposition that works for any two superimposable Christoffel words. Finally, some new properties of Christoffel words are obtained as well as a geometric proof of a classic result concerning the money problem, using Christoffel words.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
