An approximation theorem for entire functions of exponential type and stability of zero sequences

Let $L$ be an entire function of exponential type in $\mathbb C$ with indicator function $h_L$; let $\Lambda=\{\lambda_n\}$, $n=1,2,\dots$, be a subsequence of zeros of the entire function of exponential type $L\not\equiv0$; let $\Gamma=\{\gamma_n\}$ be a complex number sequence and assume that
$$ \sum_n\biggl|\frac1{\lambda_n}-\frac1{\gamma_n}\biggr|<\infty. $$

A simple construction of a sequence of entire functions of exponential type $\{L_n\}$ transforming $\Lambda$ into a subsequence $\Gamma$ of zeros of an entire function of exponential type $G\not\equiv0$ such that $h_G=h_L$ is put forward (an approximation theorem). This result is applied to stability problems of zero sequences and non-uniqueness sequences for spaces of entire functions of exponential type with constraints on the indicators and to the problem of the stability of the completeness property of exponential systems in the space of germs of analytic functions on a compact convex set.