On the factorization of infinitely divisible distributions

The article deals with the necessary and sufficient condition under which the infinitely divisible law $F$ having the finite spectral Levy's measure $\mu$ which is concentrated on the set of positive rational numbers and
$$ \mu([y,\infty))=O\{\exp(-Ky^2)\},\quad y\to+\infty,\quad\exists K>0, $$
belongs to $I_0$. The following result is also established: if $F\in I_0$ and $(\alpha\in(0,1))$
$$ \varliminf_{r\to0}\ln\mu([\alpha r,r])/\ln(1/|r|)>1, $$
then $F$ belongs to Linnik's class $\mathfrak E$.