Vanishing of an exponential sum $f(x)=\sum_{k=1}^n a_k e^{ik b_k}$ on sets of the form $\{ \alpha l : l = -m, \dots, m \}$

Question

Suppose that $f(x)=\sum_{k=1}^n a_k e^{ik b_k}$ is an exponential sum with frequencies $b_k \in \mathbb R$ and coefficients $a_k \in \mathbb C$. Further, let $S(\alpha, m)$ be the set $$ S(\alpha, m) = \{ \alpha l : l = -m, \dots, m-1, m \} \subset \mathbb R, \quad \alpha >0, \quad m \in \mathbb N. $$ Assuming that $N$ is fixed, can I find $\alpha_1, \alpha_2,m$ such that if $f$ vanishes on $S(\alpha_1, m) \cup S(\alpha_1, m)$ then $f$ vanishes everywhere on the real line? The point is here that I fix $N$ but I want to choose $\alpha_1, \alpha_2,m$ independent of the coefficients $a_k$ and frequencies $b_k$.