A construction of complete minimal, but not uniformly minimal, exponential systems with real separable spectrum in $L^p$ and $C$

We construct real separable sequences $\{\lambda_n\}$ such that the corresponding systems of exponentials $\exp(i\lambda_nt)$ are complete and minimal, but not uniformly minimal, in the spaces $L^1(-\pi,\pi)$, $L^p(-\pi,\pi)$, $1\le p<\infty$, $C[-\pi,\pi]$.