On some classes of infinitely divisible laws

The paper establishes sufficient conditions under which a probability distribution belongs to the class $I_0$ of Ju. V. Linnik. These conditions are of qualitatively new type and imply, in particular, that an arbitrary perfect set on the real line, with the origin excluded, occurs as the Poisson spectrum of a law from the class $I_0$. It is furthermore shown that in the class of all infinitely divisible laws, the laws from $I_0$ form an everywhere dense set relative to the Lévy metric.