Let $\lambda_n = n + \delta_n $ for all $n \in \mathbb{Z}$ where $\delta_n$ are a sequence of real numbers in $\ell^2(\mathbb{Z})$. I am looking to prove that the sequence $(x_n)_{n \in \mathbb{Z}} = (e^{i \lambda_n t})_{n \in \mathbb{Z}}$ is a minimal sequence in $L^2([-\pi, \pi])$, in the sense that
\begin{equation} \forall n \in \mathbb{Z}, \quad \|x_n - x \|_{L^2([-\pi, \pi])} > 0 \quad \forall x \in \overline{\text{span} \{x_k, k \in \mathbb{Z} \setminus \{n \} \}}. \end{equation}
I haven't really broken through aside from writing out the integral, so I would appreciate any input.
Edit: Assume moreover that $\delta_n \neq \delta_k $ for every $n \neq k$ (thanks to mechanodroid).
Edit: My question has gotten an answer at MO (https://mathoverflow.net/questions/289763/a-minimal-sequence-in-l2-pi-pi), but I leave it open if someone wants to add a suggestion.