Completeness and Basis Properties of Systems of Exponentials in Weighted Spaces $L^p(-\pi,\pi)$

We consider the system of exponentials $e(\Lambda)=\{e^{i\lambda_nt}\}_{n\in\mathbb Z}$, where
$$ \lambda_n=n+\biggl(\frac{1+\alpha}p+l(|n|)\biggr)\operatorname{sign}n, $$
$l(t)$ is a slowly varying function, and $l(t)\to 0$, $t\to\infty$. We obtain an estimate for the generating function of the sequence $\{\lambda_n\}$ and, with its help, find a completeness criterion and a basis condition for the system $e(\Lambda)$ in the weight spaces $L^p(-\pi,\pi)$. We also study some special cases of the function $l(t)$.