Shifts of a sequence of integers that generate functions invertible in the sense of Ehrenpreis

Consider the Schwartz algebra $\mathcal P$, which consists of all entire functions of exponential type and polynomial growth along the real axis. An element $f$ of $\mathcal P$ is said to be invertible in the sense of Ehrenpreis if the principal ideal generated by $f$ is closed. It is clear that the sequence of integers is the zero set of an Ehrenpreis invertible function. For a given unbounded function $l(t)$ on the nonnegative semi-axis, restrictions are studied under which the perturbed sequence $\{k+l(|k|)\}$, $k=\pm 1$, $\pm 2,\dots,$ is a zero set of an Ehrenpreis invertible function.