AI 中文总结
研究循环群\(\mathbb{Z}/N\mathbb{Z}\)上多重集\(S\)相关的余弦和，借助单位根消失和等理论得出余弦和消失准则及傅里叶刚性，应用于循环凯莱图得到零特征值等相关结果。
AI 中文摘要
对于循环群\(\mathbb{Z}/N\mathbb{Z}\)上的多重集\(S\)，我们通过将与\(S\)相关的有理角余弦函数的有限和转化为群环\(\mathbb{Z}[\mathbb{Z}/N\mathbb{Z}]\)中元素的求值来研究。利用单位根的消失和，特别是Lam-Leung理论，我们得到了某些条件下余弦和消失的准则，并证明了一个小权重的傅里叶刚性。然后我们将这些代数结果应用于循环凯莱图，得出零特征值准则、小支撑情况下非零特征值的重数界，以及生成集是单位群子群的无平方因子情况的描述。
英文摘要
For a multiset $S$ on the cyclic group $\mathbb{Z}/N\mathbb{Z}$, we study finite sums of cosine functions of rational angles associated to $S$ by translating them as evaluations of elements in the group ring $\mathbb{Z}[\mathbb{Z}/N\mathbb{Z}]$. Using vanishing sums of roots of unity, especially the Lam-Leung theory, we obtain criteria for the vanishing of the cosine sums under some conditions, and prove a small-weight Fourier rigidity. We then apply these algebraic results to cyclic Cayley graphs, deriving the zero-eigenvalue criteria, multiplicity bounds for nonzero eigenvalues in the small-support case, and a description of the square-free case where the generating set is a subgroup of the unit group.
Comments32 pages
