Let $\lambda\_n = n + \delta\_n $ for all $n \in \mathbb{Z}$ where $\delta\_n$ are a sequence of real numbers in $\ell^2(\mathbb{Z})$. How can one show that the sequence $(x\_n)\_{n \in \mathbb{Z}} = (e^{i \lambda\_n t})\_{n \in \mathbb{Z}}$ is minimal 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}



A minimal sequence in $L^2([-\pi, \pi])$
First, you need to say what $\delta\_n$'s are. 


I am assuming you meant $0\le \delta\_n\le 1$. This and somewhat more general statement follows from a formula of Carleman: if $\liminf \frac{n}{\lambda\_n}>\frac{A}{\pi}$, then the system is complete in $L\_2[-A,A]$. For detailed proof see the chapter 3 of [Young's book](https://books.google.com/books/about/An_Introduction_to_Nonharmonic_Fourier_S.html?id=81pFYoT4keEC&printsec=frontcover&source=kp_read_button#v=onepage&q&f=false)


