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arXiv · 2607.20907

On structured cosine sums and applications

Qin Xue
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Abstract

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.

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Qin Xue. 2026-07-23. On structured cosine sums and applications. https://arxiv.org/abs/2607.20907

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