The structure of infinitely divisible distributions on a bicompact Abelian group

Any probability distribution can be written in the form
$$ F=\alpha_1F_1+\alpha_2F_2+\alpha_3F_3,\quad\alpha_j\ge0,\quad\alpha_1+\alpha_2+\alpha_3=1, $$
where $F_1$ is an absolutely continuous, $F_2$ a singular and $F_3$ a discrete probability distribution.
We consider the following problem: what properties of the spectral measure of an infinitely divisible distribution $F$ involve $\alpha_j>0$ ($j=1,2,3$)?