Bernoulli convolutions associated with certain non—Pisot numbers

Abstract

The Bernoulli convolution νλ measure is shown to be absolutely continuous with L2 density for almost all 1/2<λ<1, and singular if λ- is a Pisot number. Itis an open question whether the Pisot type Bernoulli convolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convolutions νλ such that their density functions, if they exist, are not L2. We also construct other Bernolulli convolutions whose density functions, if they exist, behave rather badly.