Consider the lattice in \(\mathbb{R}^{d}\) given by \[\begin{align} \Lambda(\theta_1,\dots,\theta_{d-1})= \begin{bmatrix} \omega & & & & & \\ & \omega & & & & \\ & & \omega & & & \\ & & & \ddots & \\ \theta_1 & \theta_2 & \dots & \theta_{d-2} & \theta_{d-1} \omega^{d-1} \end{bmatrix} \mathbb{Z}^{d}. \end{align}\]

Then, we have that

\[\begin{equation} \int_{ \theta \in [0,1]^{d-1}} \sum_{x \in \Lambda(\theta)}f(x) = f(0) + \int_{\mathbb{R}^{d}} f(x) dx. \end{equation}\]