Bernoulli convolutions associated with certain non-Pisot numbers

Abstract

The Bernoulli convolution ν λ measure is shown to be absolutely continuous with L 2 density for almost all 1 2 <λ<1 , and singular if λ −1 is a Pisot number. It is 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 ν λ such that their density functions, if they exist, are not L 2. We also construct other Bernoulli convolutions whose density functions, if they exist, behave rather badly.