On a class of $N$-dimensional trigonometric series

An analog of Fomin's well-known one-dimensional theorem is proved for trigonometric series of the form
$$ \lambda_0+\sum_{l=1}^\infty\lambda_l\sum_{k\in lV\setminus(l-1)V}e^{ikx}, \qquad \lambda_l\to0 \quad\text{as}\quad l\to\infty, $$
given on an $N$-dimensional torus, where $V$ is some polyhedron in $\mathbb R^N$.