A necessary condition for the uniform minimality of a system of exponentials in $L^p$ spaces on the line

A necessary condition for the uniform minimality of a system of weighted exponentials
$$ \exp(-i\lambda_nt-a|t|^\alpha), \qquad a>0, \quad \alpha >1, $$
is obtained in the spaces $L^p$ $(1\leqslant p<\infty)$ and $C_0$ on the real line and the half-line. This condition is stated in terms of the indicator of the entire function of order $\beta=\alpha/(\alpha-1)$ with zero set coinciding with the sequence $\lambda_n$. This condition is used to show that there are no bases among the known complete minimal systems of this form in the above-indicated spaces.