AI 中文总结
本研究推广堆叠与双滑结构为交替结构，强化w-循环定理及相关结果，证明一般单词存在交替双滑结构，解决三个猜想（对一般单词成立）并为前两个找到反例。
AI 中文摘要
近期两篇相关论文[arXiv:2412.13941、arXiv:2409.03626]建立了两类主题的联系：一是w-循环定理（关于在特定图中读出单词w的循环出现次数的定理），二是单词测度上的特征期望。w-循环定理分别被[arXiv:1410.2540]通过堆叠法、[arXiv:1410.2579]通过双滑结构独立证明。本研究将堆叠法与双滑结构推广为交替堆叠法与交替双滑结构，证明这能显著强化适用于此类交替结构的单词的w-循环定理，进而强化上述两篇近期论文的结果。还证明一般单词均存在交替双滑结构，故强化后的结果适用于一般单词。利用新方法，本研究解决了Wilton、Hanany-Puder及Puder-Shomroni的猜想，证明这三个猜想对一般单词均成立，但也为前两个猜想找到了反例。
英文摘要
Recently, two related papers [arXiv:2412.13941, arXiv:2409.03626] found a connection between two subjects: the w-cycle theorem, which is a theorem about counting appearances of cycles reading out a word w in certain graphs, and character expectations on word measures. The w-cycle theorem was proven independently by [arXiv:1410.2540] using stackings and by [arXiv:1410.2579] using bislim structures. In the current work, we generalize stackings and bislim structures to alternating stackings and alternating bislim structures. We show how this significantly strengthens the w-cycle theorem for words admitting such alternating structures, and as a result, also strengthens the recent results of [arXiv:2412.13941] and [arXiv:2409.03626]. We show that generic words admit alternating bislim structures, and therefore, the strengthened results hold for generic words. Using our new machinery, we address conjectures of Wilton, of Hanany-Puder and of Puder-Shomroni. We prove that all three conjectures hold for generic words, but we also find counterexamples for the first two.
Comments59 pages, 22 figures
