Zeros of the Densities of Infinitely Divisible Measures

We consider an infinitely divisible measure $\mu$ on a locally compact Abelian group. If $\mu \ll \lambda$ (Haar measure), and if the semigroup generated by the support of the corresponding Levy measure $\nu$ is the closure of an angular semigroup, then $\mu \sim \lambda$ over the support of $\mu$. In particular, if $\int|\chi(x) - 1|\nu(dx) < \infty$, for all characters $\chi$, or if $\nu \ll \lambda$ then $\mu \ll \lambda$ implies $\mu \sim \lambda$ over the support of $\mu$.