A Cauchy estimate for an exponential polynomial that vanishes on a grid

Let $y_1, \dots, y_l \in \mathbb{C}$ be linearly independent over $\mathbb{Q}$, and let $F(z) = \sum_{k \in s} c_k z^{e_k} e^{\omega_k z}$ with $|\omega_k| \le W$ and $e_k \le E$ for all $k$. Let $t_j \le A_j$ be natural numbers, and suppose that $F$ vanishes to order at least $n$ at every grid point $\sum_j m'_j y_j$ with $m'_j < t_j$. Let $u \ge 2$ and $Z \ge \big(\sum_j A_j |y_j| + 1\big)(u + 1)$. Then at every point $w = \sum_j m_j y_j$ with $m_j < A_j$, and for every $r$,

This is the analytic step of the extrapolation arguments: the $\prod_j t_j$ grid points are distinct because the $y_j$ are $\mathbb{Q}$-linearly independent, and the Cauchy estimate with zeros (FourExp.cauchy_estimate_with_zeros) is applied on a circle of radius $(\rho + 1)u$ about $w$, where $\rho = \sum_j A_j |y_j|$.