Absolute continuity of some semi-selfdecomposable distributions and self-similar measures

P.S. numbers are introduced in relation to absolute continuity of some infinite Bernoulli convolutions. Absolute continuity and continuous singularity of some semi-selfdecomposable distributions are studied as marginal distributions of subordinators. It is shown that these properties are widely different according as their spans are P.V. numbers or the reciprocals of P.S. numbers. A simple example of a subordinator whose distribution is continuous singular for small time and absolutely continuous for large time is given. Absolute continuity of convolutions of multidimensional homogeneous self-similar measures are also discussed. Received: 27 December 1998 / Revised version: 20 August 1999 / Published online: 31 May 2000