Discrete tori and trigonometric sums (Q2082309)

For \(v=(v_1,\cdots, v_n)\in \mathbb{Z}^n\) set \(|v|=|v_1|+\cdots +|v_n|\) and \(\bar{v}=(|v_1|,\cdots, |v_n|)\). For \(v\in \mathbb{Z}^n_+\) set \(v!=v_1!\cdots v_n!\). Fix an integer valued \(n\times n\) matrix \(M\) with \(m:=\text{det}M>1\). Let \(M^*\) be the transpose of \(M\). For any non-negative integer \(s\) set \[ C_s(M)=\sum_{v\in M\mathbb{Z}^n, z\in \mathbb{Z}^n_+, |v|+2|z|=s} \frac{s!}{z! (\bar{v}+z)!}. \] The main result of this paper states that for the torus \[ W=(M^*)^{-1}\mathbb{Z}^n/\mathbb{Z}^n \] and for any non-negative integer \(s\), it holds that \[ \sum_{w\in W}\left(\sum_{k=1}^n \cos 2\pi w_k \right)^s=\frac{m}{2^s}C_s(M). \]